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arXiv:1308.0587v1 [hep-ex] 2 Aug 2013

TEL AVIV UNIVERSITY

RAYMOND AND BEVERLY SACKLER FACULTY OF EXACT SCIENCES

SCHOOL OF PHYSICS AND ASTRONOMY

Double parton scattering in four-jet events

in pp collisions at √s = 7 TeV with the

ATLAS experiment at the LHC

Thesis submitted to the senate of Tel Aviv University

towards the degree “Doctor of Philosophy”

by

Iftach Sadeh

January 2013

Thesis supervisors,

Prof. Halina Abramowicz and

Prof. Aharon Levy.

The research for this thesis has been carried out in

the Particle Physics Department of Tel Aviv University, under

the supervision of Prof. Halina Abramowicz and Prof. Aharon Levy.

2

Abstract

The dijet double-differential cross section is measured as a function of the dijet invariant mass, using

data taken during 2010 and during 2011 with the ATLAS experiment at the LHC, with a center-of-

mass energy, √s = 7 TeV. The measurements are sensitive to invariant masses between 70 GeV

and 4.27 TeV with center-of-mass jet rapidities up to 3.5. A novel technique to correct jets for

pile-up (additional proton-proton collisions) in the 2011 data is developed and subsequently used in

the measurement. The data are found to be consistent over 12 orders of magnitude with fixed-order

NLO pQCD predictions provided by NLOJET++. The results constitute a stringent test of pQCD,

in an energy regime previously unexplored.

The dijet analysis is a confidence building step for the extraction of the signal of hard dou-

ble parton scattering in four-jet events, and subsequent extraction of the effective overlap area be-

tween the interacting protons, expressed in terms of the variable, σeff. The measurement of double

parton scattering is performed using the 2010 ATLAS data. The rate of double parton scatter-

ing events is estimated using a neural network. A clear signal is observed, under the assumption

that the double parton scattering signal can be represented by a random combination of exclu-

sive dijet production. The fraction of double parton scattering candidate events is determined to be

fDPS = 0.081 ± 0.004 (stat.) + 0.025

−0.014

(syst.) in the analyzed phase-space of four-jet topologies. Com-

bined with the measurement of the dijet and four-jet cross sections in the appropriate phase-space

regions, the effective cross section is found to be σeff = 16.0 + 0.5

−0.8

(stat.) + 1.9

−3.5

(syst.) mb. This result

is consistent within the quoted uncertainties with previous measurements of σeff at center-of-mass

energies between 63 GeV and 7 TeV, using several final states.

3

Acknowledgements

It is with immense gratitude that I acknowledge my supervisors, Prof. Halina Abramowicz and

Prof. Aharon Levy, not only for teaching me physics, but for making me feel like part of the family.

This thesis is a tribute to their steadfast encouragement, guidance and support, making the course

of my studies a pleasure which is difficult to leave behind. I could not have hoped for a better pair

of physicists, or for a more caring pair of people, to be a student of.

I consider it an honor to have worked on the double parton scattering analysis with my colleague

and friend, Orel Gueta. Defeats and victories, late night insights and countless discussions, have all

culminated in a measurement to be proud of. Great thanks is also owed to Dr. Arthur Moraes, for

inspiring us with his ideas about multiple interactions, and for keeping faith when all seemed lost.

In addition, the shared experience of Dr. Frank Krauss and of Dr. Eleanor Dobson has been instru-

mental in the success of the analysis, and is much appreciated. On the theoretical side of things, the

fruitful discussions with Prof. Yuri Dokshitzer, Prof. Mark Strikman, Prof. Leonid Frankfurt and

Prof. Evgeny Levin were of great help in formulating the strategy of the analysis.

I would like to show special appreciation to Dr. Sven Menke and Dr. Teresa Barillari for the

stimulating discussions on all things calorimeter (and then some), and to Prof. Allen Caldwell in

addition. They have made my visits to the Max-Planck-Institut für Physik in Munich an experience

to remember, intellectually and otherwise.

Several people have contributed greatly to my understanding of the workings of the ATLAS

experiment, be it in terms of performance or of physics. The debates with Dr. Ariel Schwartz-

man and Dr. Peter Loch have been of great help in my development of a pile-up correction using

the jet area/median method. To start me off on the invariant mass measurement, the guidance of

Dr. Christopher Meyer via many hours of discussions and a multitude of emails, has been invalu-

able. I would like to thank Dr. Pavol Strizenec for starting me off on ATLAS software. I would also

like to Acknowledge Dr. Eric Feng and Dr. Serguei Yanush, for developing the software framework

for calculation of the non-perturbative corrections to the invariant mass theoretical predictions.

I am grateful to my colleagues and the members of the group at Tel Aviv University, Prof. Gideon

Alexander, Prof. Erez Etzion, Prof. Abner Soffer, Dr. Gideon Bella, Dr. Sergey Kananov, Dr. Zhenya

Gurvich, Dr. Ronen Ingbir, Amir Stern, Rina Schwartz and Itamar Levy, for allowing me to bounce

ideas off of, for the stimulating discussions, and for the pleasure of their company.

Finally, completing this thesis would not have been possible without the love and support of my

family. As with everything else which is good in my life, it all starts and ends with them. Thank

you for being you.

4

Preface

The goal of this thesis is the study of double-parton scattering (DPS) in four-jet events with the

ATLAS experiment. In order to extract DPS in this channel, a good understanding of the reconstruc-

tion and calibration of jets is needed. A comprehensive framework exists in ATLAS for this purpose,

featuring two main calibration schemes, referred to as the electromagnetic (EM) and the local-hadron

(LCW) calibrations. These rely on extensive test-beam and simulation campaigns, which are the re-

sult of the efforts of a large number of individual researchers and analysis subgroups.

Test-beam data taken during 2004 served to test the detector performance and to validate the

description of the data by simulations. Due to changing software models, these data became incom-

patible with current ATLAS reconstruction tools. In order to maintain future access, the information

had to be made persistent, i.e., compatible with all future software. The first project which had been

undertaken by the author of this thesis in ATLAS, was persistification of the test-beam data. This

was followed by continued support and maintenance of the ATLAS calorimeter reconstruction soft-

ware, as part of the operational contribution of the author to the experiment.

One of the major challenges in the calibration of jets in ATLAS, is the existence of pile-up,

additional proton-proton (pp) collisions, which coincide with the hard scattering of interest. The

effects of pile-up on final states which involve jets are complicated. Pile-up tends to both bias

the energy of jets which originate from the hard interaction, and to introduce additional jets which

originate from the extraneous pp collisions. The current pile-up subtraction method in ATLAS in-

volves a simulation-based scheme; it affects an average correction for jet energies, based on the

instantaneous luminosity and on the number of reconstructed vertices in an event. An alternate,

event-by-event-based correction, has been developed by the author for the LCW calibration scheme.

In the new correction, referred to as the jet area/median method, the area of a given jet and the

“local” energy-density, are used in order to subtract pile-up energy from the jet. The method takes

advantage both of the average response of the calorimeter to pile-up energy, and of the observed

energy in the vicinity of the jet of interest. The median method is completely data-driven. Con-

sequently, compared to the nominal pile-up correction, the uncertainties on the energy correction

associated with the simulation of pile-up were reduced.

The new pile-up correction was developed by the author, initially by using the ATLAS detector

simulation. It was subsequently validated by the author with the 2011 ATLAS dataset, in which

the rate of pile-up is high. The validation included in-situ measurements of several observables,

one of which was the invariant mass of dijets, the system of the two jets with the highest transverse

momentum in an event. The author also performed measurements of the dijet double-differential

cross section for different center-of-mass jet rapidities, as a function of the invariant mass of dijets.

The invariant mass spectra had previously been measured in ATLAS using the 2010 data. The new

measurements, performed for the first time on the 2011 data, were found to be compatible with

the previous observations, showing that the pile-up corrections were under control. In addition, the

measurement using the larger dataset recorded during 2011, served to extend the experimental reach

to higher values of the invariant mass.

5

Using the improved jet calibration, events with four-jets in the final state were investigated as part

of the DPS analysis. It was shown by the author that the deterioration of the energy resolution due

to pile-up distorted greatly the observables of the measurement. The scope of the measurement had,

therefore, to be limited. As this is a first measurement of double parton scattering in this channel in

ATLAS, it was decided to choose a conservative approach and to limit the systematic uncertainties

as much as possible. The analysis, therefore, used only single-vertex events from the 2010 dataset,

for which pile-up corrections are small.

Several strategies of the DPS analysis were explored by the author. For instance, the author at-

tempted to measure the fraction of DPS events, by exploiting the sum of the pair-wise transverse

momentum balance between different jet-pair combinations in a four-jet event, as previously done

by the CDF collaboration. The problem with this type of approach was the absence of an appro-

priate simulation sample, in which DPS events within the required phase-space were generated. In

order to avoid dependence on simulated DPS events, and in addition, to increase the robustness of

the analysis, the author decided to utilize a neural network. Two input samples were prepared by the

author for the neural network. The first consisted of simulated events in which multiple-interactions

were switched off. The second, which stood for the DPS signal, was comprised of overlaid simu-

lated dijet events. Utilizing the neural network, the fraction of double parton scattering events was

extracted from the data, and the effective cross section for DPS was subsequently measured. The

result was found to be compatible with previous measurements which had used several final states,

at center-of-mass energies between 63 GeV and 7 TeV.

6

Table of Contents

Table of Contents

7

1 Introduction

9

2 Theoretical background

13

2.1 The Standard Model of particle physics . . . . . . . . . . . . . . . . . . . . . . . .

13

2.2 QuantumChromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.3 Full description of a hard pp collision . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3.1 Thehardscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.3.2 Non-perturbativeeffects . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.4 Doublepartonscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

3 The ATLAS experiment at the LHC

29

3.1 TheLargeHadronCollider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

3.2 The ATLAS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

4 Monte Carlo simulation

33

4.1 Event generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

4.2 SimulationoftheATLASdetector . . . . . . . . . . . . . . . . . . . . . . . . . .

34

4.3 Bunchtrainstructureandoverlappingevents . . . . . . . . . . . . . . . . . . . . .

35

5 Jet reconstruction and calibration

37

5.1 Jetreconstructionalgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

5.2 Inputstojetreconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

5.2.1 Calorimeterjets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

5.2.2 Othertypesofjets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

5.3 Jetenergycalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

5.4 Jetqualityselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

5.5 Systematic uncertainties on the kinematic properties of jets . . . . . . . . . . . . .

45

5.5.1 Jet energy scale uncertainties . . . . . . . . . . . . . . . . . . . . . . . . .

45

5.5.2 Jet energy and angular resolution . . . . . . . . . . . . . . . . . . . . . . .

48

6 Data selection

53

6.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

6.2 Triggerandluminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

6.2.1 Descriptionofthetrigger . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

6.2.2 Triggerefficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

6.2.3 Luminosity calculation using a two-trigger selection scheme . . . . . . . .

56

7 The jet area/median method for pile-up subtraction

59

7

Table of Contents

7.1 Pile-upinATLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

7.1.1 In-andout-of-timepile-up . . . . . . . . . . . . . . . . . . . . . . . . . .

59

7.1.2 Thejetoffsetpile-upcorrection . . . . . . . . . . . . . . . . . . . . . . . .

62

7.2 A pile-up correction using the jet area/median method . . . . . . . . . . . . . . . .

65

7.2.1 Parametrizationoftheaveragepile-upenergy . . . . . . . . . . . . . . . .

66

7.2.2 Pile-upsubtractionwiththemedian . . . . . . . . . . . . . . . . . . . . . .

68

7.2.3 Performance of the median correction in MC . . . . . . . . . . . . . . . . .

70

7.3 Systematic checks of the median correction . . . . . . . . . . . . . . . . . . . . .

74

7.3.1 Stability in MC for different parameters of the jet algorithm . . . . . . . . .

74

7.3.2 Stabilityindata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

7.3.3 Associated systematic uncertainty of the jet energy scale . . . . . . . . . .

81

7.4 In-situmeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

7.4.1 Dijetbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

7.4.2 Jet pt anddijetinvariantmassspectra . . . . . . . . . . . . . . . . . . . .

84

7.5 Summary of the performance of the median pile-up correction . . . . . . . . . . .

86

8 Dijet mass distribution

88

8.1 Crosssectiondefinition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

8.2 TheoreticalPredictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

8.2.1 NLO calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

8.2.2 Non-perturbativecorrections . . . . . . . . . . . . . . . . . . . . . . . . .

90

8.2.3 UncertaintiesontheNLOprediction . . . . . . . . . . . . . . . . . . . . .

92

8.3 Biasandresolutionintheinvariantmass . . . . . . . . . . . . . . . . . . . . . . .

92

8.4 Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

8.5 Systematicuncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

8.5.1 Systematicuncertaintyonthejetenergyscale . . . . . . . . . . . . . . . . 101

8.5.2 Othersourcesofuncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.5.3 Totalsystematicuncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9 Hard double parton scattering in four-jet events

107

9.1 Strategyoftheanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.2 Measurement of the ratio of dijet and four-jet cross sections . . . . . . . . . . . . . 111

9.3 Extraction of the fraction of DPS events using a neural network . . . . . . . . . . . 113

9.3.1 Inputsamplestotheneuralnetwork . . . . . . . . . . . . . . . . . . . . . 113

9.3.2 Inputvariablestotheneuralnetwork . . . . . . . . . . . . . . . . . . . . . 116

9.3.3 Trainingandoutputoftheneuralnetwork . . . . . . . . . . . . . . . . . . 121

9.4 Systematicandstatisticaluncertainties . . . . . . . . . . . . . . . . . . . . . . . . 124

9.5 Resultsandsummaryofthemeasurement . . . . . . . . . . . . . . . . . . . . . . 127

10 Summary

129

A Additional figures and tables

130

A.1 Jetreconstructionandcalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

A.2 Dataselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.3 Thejetarea/medianmethodforpile-upsubtraction . . . . . . . . . . . . . . . . . 137

A.4 Dijetmassdistribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

A.5 Harddoublepartonscatteringinfour-jetevents . . . . . . . . . . . . . . . . . . . 148

Bibliography

151

8

1. Introduction

The Standard Model (SM) [1] is one of the major intellectual achievements of the twentieth cen-

tury. In the late 1960s and early 1970s, decades of path breaking experiments culminated in the

emergence of a comprehensive theory of particle physics. This theory identifies the fundamental

constituents of matter and combines the theory of electromagnetic, weak and strong interactions.

Numerous measurements at energy scales from a few eV to several TeV are reproduced by the

SM, and many of its predictions, e.g., the existence of the W and Z bosons, have been found to be

realised in nature. By now, only the source of electroweak symmetry breaking, which in the SM is

attributed to the Higgs mechanism, has not been verified experimentally. A particle with properties

consistent with those of the Higgs boson has recently been discovered at the LHC by the ATLAS [2]

and CMS [3] experiments; the final piece of the puzzle is therefore within reach.

The Standard Model falls short of being a complete theory of fundamental interactions because

it does not incorporate the full theory of gravitation, as described by general relativity; nor does

it predict the accelerating expansion of the universe, as possibly described by dark energy. The

theory does not contain any viable dark matter particle that possesses all of the required properties

deduced from observational cosmology. It also does not correctly account for neutrino oscillations

or for the non-zero neutrino masses. Although the SM is believed to be theoretically self-consistent,

it has several apparently unnatural properties, giving rise to puzzles like the strong CP problem and

the hierarchy problem [4]. The SM is therefore viewed as an effective field theory that is valid at

the lower energy scales where measurements have been performed, but which arises from a more

fundamental theory at higher scales. It is therefore expected that a more fundamental theory exists

beyond the TeV energy scale that explains the missing features.

Verification of the Higgs mechanism and the search for new physics beyond the SM are two of the

primary enterprises in particle physics today. Any experimental search for the Higgs boson or for

new interactions or particles, requires a detailed understanding of the strong interactions, described

by Quantum Chromodynamics (QCD). It is common to discuss high-energy phenomena involving

QCD in terms of partons (quarks and gluons), yet partons are never visible in their own right.

Almost immediately after being produced, a quark or gluon fragments and hadronizes, leading to

a collimated spray of energetic hadrons, which may be characterized by a jet [5]. Since partons

interact strongly, jet production is the dominant hard scattering process in the SM. Figure 1.1 shows

the production cross section for various processes as a function of the center-of-mass energy of an

accelerator. One may compare the production cross section for energetic jets with transverse energy

above 100 GeV, with those of other processes (W and Z) involving leptons. At the center-of-mass

energy of the LHC, the former is much larger than the latter.

Because of the large cross sections, jet production provides an ideal avenue to probe QCD and

parton distribution functions [6–9], which describe the distribution of the momenta of quarks and

gluons within a proton. Processes involving jets also serve as large backgrounds in many searches

for new physics. Many models predict the production of new heavy and coloured particles at the

LHC. These particles are expected to decay into quarks and gluons, which are detected as particle

jets. Such models may therefore be tested by measuring the rate of jet production and comparing it

9

1. Introduction

0.1

1

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10-7

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10-1

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101

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10-7

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109

σ

ZZ

σ

WW

σ

WH

σ

VBF

MH=125 GeV

WJS2012

σ

jet

(E

T

jet > 100 GeV)

σ

jet

(E

T

jet > √s/20)

σ

ggH

LHC

Tevatron

e

ve

n

ts / se

c fo

r

L

=

1

0

3

3

cm

-2

s-1

σ

b

σ

tot

proton - (anti)proton cross sections

σ

W

σ

Z

σ

t

σ

((((

n

b

))))

√s (TeV)

{

Figure 1.1. Cross sections for different physics processes as a function of center-of-mass energy,

√s, assuming the existance of a Higgs boson with a mass of 125 GeV. (Figure by J. Stirling, taken

from http://projects.hepforge.org/mstwpdf/.)

to the expectations from QCD [10]. Other models, such as quark compositeness (the hypothesis that

quarks are composed of more fundamental particles), may be realised through contact interactions

between quarks. Such new phenomena may be discovered by measuring the kinematic distributions

of jets [11].

In the most common final state involving jets at the LHC, two jets with high transverse momen-

tum, pt, emerge from the interaction, as illustrated in figure 1.2a. These dijet events are particularly

useful for measuring quantities associated with the initial interaction, such as the polar scattering

angle in the two-parton center-of-mass frame, and the dijet invariant mass. Precise tests of pertur-

bative QCD (pQCD) at high energies, may therefore be carried out by comparing the theoretical

predictions to the experimental distributions. In addition, new physics may manifest itself with e.g.,

the production of a new massive particle, which subsequently decays into a high-mass dijet sys-

tem [12]. Final states composed of low-pt or forward jets are also interesting. These topologies

are sensitive to QCD effects which can not be calculated using perturbation theory; as such, their

measurement may help to improve the phenomenological models which are currently in use in this

regime.

10

(a)

(b)

Figure 1.2. (a) Event display of a high-mass (4.04 TeV) dijet event taken in April 2011. The two

jets which comprise the dijet system are marked by the colours red and green; these respectively

have transverse momenta, pt = 1.85 and 1.84 TeV, pseudo-rapidities, η = 0.32 and − 0.53 and

azimuthal angles, φ = 2.2 and −0.92. Shown are a view along the beam axis (left), and the angular

distribution of transverse energy in η and φ (bottom right).

(b) Event display of an event with four reconstructed jets, taken in April 2010. The four jets all have

pt > 50 GeV, where the highest-pt jet has transverse momentum of 108 GeV. Depicted points-of-

view are along the beam axis (left), parallel to the beam axis (top right), and the angular distribution

of transverse energy in η and φ (bottom right).

One of the important sources of background for physics searches at the LHC are multiple-parton

interactions (MPI). In a generic hadron-hadron collision, several partons in one hadron scatter on

counterparts from the other hadron. Most of these interactions are soft, and so generally do not result

in high-pt jets. However, the kinematic reach of the LHC, which extends to high energies and low

fractions of proton momenta, enhances the probability of hard MPI relative to past experiments [13].

Hard MPI constitute a source of background for e.g., Higgs production [14, 15], and can possibly

influence observed rates of other final states, involving the decay of heavy objects. The existing

phenomenology of MPI is based on several simplifying assumptions. Recent interest has produced

some advancements [16–22], however, a systematic treatment within QCD remains to be developed.

The simplest case of MPI is that of double parton scattering (DPS). Measurements of DPS at

energies between 63 GeV and 1.96 TeV in the four-jet and γ + three-jets channels have previously

been performed [23–27]. In addition, a measurement using 7 TeV ATLAS data with W + two-jet

production in the final state has recently been released [28]. These have proven helpful in constrain-

ing the phenomenological models which describe DPS, and in the development of double parton

distribution functions, as e.g., discussed in [17]. An additional channel in which DPS may be mea-

sured at the LHC is four-jet production. An ATLAS event display of a four-jet event is presented

in figure 1.2b for illustration. A four-jet final state may arise due to a single parton-parton colli-

sion, accompanied by additional radiation. Alternatively, it can also originate from two separate

parton-parton collisions, each producing a pair of jets. The latter case has distinguishing kinematic

characteristics, and so the rate of DPS may be estimated on average. The DPS-rate is related to

a so-called effective cross section. The latter holds information about the transverse momentum

distributions of partons in the proton, and about the correlations between partons. A measurement

of DPS in four-jet events has been performed using a sub-sample of the 2010 ATLAS dataset, and

is presented in this thesis.

11

1. Introduction

Outline of the thesis

This thesis presents a measurement of hard double parton scattering in four-jet events. The analysis

involves measurement of the dijet and four-jet cross sections, and the extraction of the rate of double

parton scattering from the four-jet sample.

In order to measure the individual cross sections, a measurement of the inclusive dijet invariant

mass distribution is first performed. Part of the experimental challenge of measurements involving

jets in ATLAS, is the calibration of the energy of jets, as these suffer from detector background

due to multiple simultaneous pp collisions, referred to as pile-up. A novel method to reduce the

pile-up background using jet areas is developed and subsequently utilized to correct the energy of

jets. The re-calibrated jets serve as input for the dijet mass distribution analysis of the 2011 ATLAS

dataset, for which the pile-up background is severe. The invariant mass measurement serves as a

confidence building process for understanding the reconstruction of jets, handling of the trigger and

the luminosity. The measurement is also performed on the 2010 dataset, and found to be compatible

with the previously published results of ATLAS, and with the present measurement, using the 2011

data.

The 2010 dijet data-sample is also an essential element in the analysis of double parton scattering.

In spite of the good understanding of how to handle the 2011 data, the measurement of double parton

scattering is limited to the 2010 data, and even to a sub-sample of these data, which include single-

vertex events. This choice is dictated by the inherent difficulty in extracting the double parton

scattering signal, and by the shortcomings of the available simulation.

Following the short introduction given here, a theoretical overview of the Standard Model and

of QCD is presented in chapter 2. The ATLAS experiment is described in chapter 3, followed by

a summary of the physics event generators and detector simulation which are used in the analysis

in chapter 4. In chapter 5 the concept of a jet is rigorously defined, followed by a discussion of

calorimeter jets in the context of the ATLAS detector; this includes a description of the energy

calibration of jets and of the systematic uncertainties associated with the calibration. Chapter 6

details the data sample which is used in the analysis, with an emphasis on the trigger selection

procedure and on calculation of the luminosity. In chapter 7 the jet areas method to subtract pile-up

background is introduced, and the performance of the pile-up correction is estimated in simulation.

Several in-situ measurements are also used for validation of the performance in data. The new jet

energy calibration which is thus developed is used to perform a measurement of the differential dijet

invariant mass cross section in chapter 8. In chapter 9, the rate of double parton scattering events is

extracted from the data using a neural network, and the effective cross section, σeff, is measured. A

summary of the results is finally given in chapter 10.

12

2. Theoretical background

2.1. The Standard Model of particle physics

The Standard Model (SM) [29] is the most successful theory describing the properties and interac-

tions (electromagnetic, weak and strong) of the elementary particles. The SM is a gauge quantum

field theory based in the symmetry group SU(3)C ×SU(2)L ×U(1)Y, where the electroweak sector

is based in the SU(2)L ×U(1)Y group, and the strong sector is based in the SU(3)C group.

Figure 2.1. The fundamental particles

of the Standard Model, sorted according

to family, generation and mass.

Interactions in the SM occur via the exchange of in-

teger spin bosons. The mediators of the electromagnetic

and strong interactions, the photon and eight gluons re-

spectively, are massless. The weak force acts via the ex-

change of three massive bosons, the W± and the Z.

The other elementary particles in the SM are half-

integer spin fermions, six quarks and six leptons. While

both groups interact via the electroweak force, only

quarks feel the strong interaction. Electrons (e), muons

(µ) and taus (τ) are massive leptons and have electrical

charge Q = −1. Their associated neutrinos, respectively

νe, νµ and ντ, do not have electrical charge. Quarks can

be classified depending on their electrical charge; quarks

u, c and t have Q = 2/3 and quarks d, s and b have

Q = −1/3. For each particle in the SM, there is an anti-

particle with opposite quantum numbers. The fundamen-

tal particles of the Standard Model, sorted according to

family, generation and mass, are listed in figure 2.1.

The SM formalism is written for massless parti-

cles. The Higgs mechanism of spontaneous symmetry

breaking is proposed for generating non-zero boson and

fermion masses. The symmetry breaking requires the in-

troduction of a new field that leads to the existence of

a new massive boson, the Higgs boson. A particle with

properties consistent with those of the Higgs boson has

recently been discovered at the LHC by the ATLAS [2] and CMS [3] experiments.

2.2. Quantum Chromodynamics

Quantum chromodynamics (QCD) [30] is the theory of strong interactions. Its fundamental con-

stituents are quarks and gluons, which are confined in the nucleon but act as free at sufficiently

13

2. Theoretical background

small scales (and high energies). The latter behaviour is called asymptotic freedom. The direct con-

sequence of confinement is that free quarks and gluons are never observed experimentally, and their

final state is a collimated shower of hadrons.

The development of QCD was posterior to that of quantum electrodynamics (QED); while the

latter was highly successful in the mid-Sixties, no information about the components of the nucleus

was available. Strong interactions were commonly described using general principles and the ex-

change of mesons [31], although the basis for theories that could eventually accommodate QCD had

also been developed [32]. A framework called the Eightfold Way [33] had been developed to orga-

nize subatomic baryons and mesons into octets. Its connection to an underlying point-like structure

of hadrons came after the so-called heroic age of deep inelastic scattering (DIS) measurements,

interpreted using the parton model [34]. These experiments and subsequent interpretations showed

that the probes scattered against point-like, spin 1/2 constituents of the nucleons that are the quarks.

The presence of spin 1 gluons was also inferred using kinematic considerations in terms of the total

momentum shared by the quarks. The QCD equivalent of the electromagnetic charge is the colour

charge; (anti-)quarks can take three (anti-)colours (red, green and blue, and their counterparts);

the eight interacting gluons exist in a superposition of colour and anti-colour states. SU(3) QCD

was established as a theoretical framework for strong interactions only following the discovery of

asymptotic freedom as a consequence of the renormalisability of the theory [35]. A short overview

of these concepts follows.

The lagrangian of QCD, confinement and asymptotic freedom

QCD is the renormalizable gauge field theory that describes the strong interaction between colored

particles in the SM, based in the SU(3) symmetric group. The lagrangian of QCD is

LQCD = ∑

q

¯ψq(iγµ

Dµ −mq) ψq −

1

4

G

A

αβ G

αβ

A ,

(2.1)

where quarks and anti-quark fields are respectively denoted by ψq and ¯ψq, quark mass is denoted

by mq, γµ

are the Dirac matrices, and Dµ stands for a covariant derivative. The sum runs over the

six different flavors of quarks. The gauge invariant gluonic field strength tensor,

GA

αβ

= [∂αGA

β −∂β GA

α −gs fABCGB

αGC

β

] ,

(2.2)

is derived from the gluon fields, GA

α, where gs is a coupling constant, fABC are the structure constants

of SU(3), and the indices A, B and C run over the eight color degrees of freedom of the gluon field.

The third term originates from the non-abelian character of the SU(3) group. It is responsible for

the gluon self-interaction, giving rise to triple and quadruple gluon vertexes. This leads to a strong

coupling, αs = g2

s /4π, that is large at low energies and small at high energies (discussed further in

the next section). Two consequences follow:

- confinement - the color field potential increases linearly with distance. Quarks and glu-

ons can never be observed as free particles, and are always inside hadrons, either as mesons

(quark-antiquark) or as baryons (three quarks, each with a different color). If two quarks

separate far enough, the field energy increases and new quarks are created, forming colorless

hadrons;

- asymptotic freedom - at small distances the strength of the strong coupling is low, such

that quark and gluons behave as if they were free. This allows the perturbative approach to be

used in the regime where αs ≪ 1.

14

2.2. Quantum Chromodynamics

Confinement and asymptotic freedom have relevant experimental consequences; quarks and glu-

ons require interactions with high energy probes to be ejected from the nucleon, and they cannot

be observed directly. What one detects instead of quarks and gluons are jets, which are collimated

showers of particles. These particles are the product of a series of steps [36],

- two hadrons collide with a large momentum transfer;

- two incoming partons from the hadrons collide and produce the hard process; at leading order

this is a 2 → 2 process, for which the cross section is calculable in pQCD. It is assumed

that the probabilities of finding partons in the proton, given by the parton density functions

(PDFs), are know;

- additional semi-hard or hard interactions may occur between the remaining partons. These

arise naturally in that the integrated cross section for hard scattering diverges for low trans-

verse momentum; the cross section becomes bigger than the total cross section, unless multi-

parton interactions (MPI) are invoked. The remaining partons are referred to as outgoing

partons;

- the incoming and outgoing partons may radiate, due to their having electromagnetic and color

charges, as described using pQCD;

- when partons are sufficiently “far” from each other, hadronisation takes place. That is, con-

finement comes into play and additional color charges (quark/anti-quark pairs) are created

such that all free partons combine into hadrons. Many of these hadrons are unstable and de-

cay further at various timescales; they or their decay products may be observed in a detector.

The products of the collision that are not directly identified with the hard scattering (hadron rem-

nants, products of soft multiple parton interactions and radiation) are conventionally defined as the

underlying event (UE).

Each of the steps described here is subject to modeling, and is accompanied by uncertainties that

are hard to quantify. Various Monte Carlo (MC) generators [37–41] adopt different approaches, in

particular for the treatment of MPI and the UE. A detailed discussion is given in section 2.3.

Renormalisation

Gluons in QCD are massless, the theory therefore results in divergences of theoretical cross section

calculations. A renormalisation procedure is necessary in order to allow the theory to give meaning-

ful (non infinite) results that can be compared to experimental measurements. This is achieved by

effectively subtracting these infinities through counter-terms embedded in so-called bare parameters

that are not measurable. The renormalisation procedure introduces a correction to the renormalised

parameter, depending on the renormalisation scale, µR, (interpreted as the scale at which the sub-

traction is made), and on the physical scale at which the measurement is made; the latter is taken

as the squared momentum transfer, Q2, in the following. Imposing the independence of the final

result in all orders of perturbation theory from the renormalisation scale, allows one to derive an ex-

plicit form for the renormalised parameter. As an example of a renormalised parameter, the strong

coupling constant in the one loop approximation (first order in perturbation theory) is

αs (Q

2) =

αs (µ2

R)

1+(β1/4π) αs (µ2

R) ln(Q2/µ2

R) =

4π

β1 ln(Q2/Λ2

QCD) ,

(2.3)

where

ΛQCD = µR exp(−

2π

β1αs (µ2

R)

)

(2.4)

15

2. Theoretical background

sets the scale of the coupling. The parameter β1 is a constant, computed by Gross Wilczek and

Politzer [42–44]; it is the first term in the beta function of the strong coupling constant,

β (αs) = −√4παs(µ2)(

αs

4π

β1 +(

αs

4π )

2

β2 +...) ,

(2.5)

which encodes the dependence of αs on the energy scale.

The coupling constant, initially scale-invariant, becomes a function of the scale of the process,

commonly referred to as a running coupling constant. The current theoretical and experimental

results for the running αs [45] are shown in figure 2.2. Contrary to QED, where the coupling

Figure 2.2. Compilation of measurements of the coupling constant, αs, as a function of the

energy scale, Q. The degree of QCD perturbation theory used in the extraction of αs is indicated in

parenthesis in the figure. (Figure taken from [45].)

constant increases with the scale of the process, gluon self-interactions result in negative values of

β (αs). The coupling constant therefore is sizeable at low values of Q2, leading to confined partons,

and decreases as Q2 increases, leading to asymptotic freedom [46].

The parton model, parton distribution functions and evolution

Asymptotic freedom allows QCD to be described using point-like constituents at sufficiently large

energies. The first evidence of this behaviour was given by the SLAC experiments [47] and inter-

preted by Feynman using the parton model [34]. Later on, it was realised that the momentum scale

16

2.2. Quantum Chromodynamics

introduced by renormalisation needed to be accommodated, and the improved parton model was

developed [48]. Starting from these ideas, the perturbative evolution of quarks and gluons can be

predicted independently of the soft, non-perturbative physics, allowing for theoretical calculation of

QCD processes.

The differential cross section for lepton-hadron (lh) inelastic scattering can be parameterised,

starting from that of elastic scattering of fundamental particles,

d2σlh

dxdQ2

=

1

q4 (f(y)xF1 (x,Q2)+g(y)F2 (x,Q2)) ,

(2.6)

where F1 and F2 are the structure functions, which reflect the structure of the nucleon. They are

parametrized in terms of the momentum transfer, Q2, and of x, which represents (at leading order)

the fraction of hadron momentum carried by the massless struck quark. Similarly, f(y) and g(y)

are functions which depend on the kinematics of the scattering, where the parameter y measures the

ratio of the energy transferred to the hadronic system, to the total leptonic energy available in the

target rest frame.

The structure function F2 can be written as

F2 =

Nq

∑

i

e2

i xfi(x) ,

(2.7)

where e2

i and fi(x) are respectively the squared charge and momentum distribution of the ith quark,

and the sum goes over all quarks in the hadron (in total Nq quarks). The fi(x) functions can be

interpreted (at leading order of perturbation theory) as the probability densities of finding a quark

with flavour i, carrying a fraction x of the hadron momentum. The momentum distribution for

a given quark or gluon is also called the parton distribution function (PDF). The two structure

functions are not independent, due to the fact that quarks have spin 1/2. This is expressed by the

Callan-Gross relation [49],

F2 = 2xF1 .

(2.8)

Probability conservation requires that

∫ 1

0

x

Nq

∑

i

fi(x)dx = 1 ,

(2.9)

also called the momentum sum rule. However, calculation of the integral comes up to about 0.5.

This calls for the presence of gluons. Conventionally, partons composing a hadron are divided

between gluons, valence quarks and sea quarks. Valence quarks are responsible for the quantum

numbers of the hadron, while sea quarks are quark/anti-quark pairs that are generated due to quan-

tum fluctuations.

The independence of the structure function from momentum transfer, Q2, at fixed values of x,

is known as Bjorken scaling. Interactions among the partons lead to deviation from the naive par-

ton model in terms of scaling violations. The latter have been observed experimentally, as seen in

figure 2.3, where measurements at a given value of x are shown to depend on Q2. Intuitively, an

increase in Q2 can be seen as an increase in the resolving power of the probe; if the internal struc-

ture of the hadron can be probed at smaller distances, then the number of partons “seen” increases,

as gluons can produce quark/anti-quark pairs and quarks or anti-quarks can radiate gluons. The

17

2. Theoretical background

❍  ✁✂✄ ☎✆✝✞

① ✟ ✠✡✠✠✠✠☛☞ ✌✟✍✎

① ✟ ✠✡✠✠✠✠✏☞ ✌✟✍✠

① ✟ ✠✡✠✠✠✎✑☞ ✌✟✎✒

① ✟ ✠✡✠✠✠✍✠☞ ✌✟✎✏

① ✟ ✠✡✠✠✠✑✍☞ ✌✟✎✓

① ✟ ✠✡✠✠✠☛☞ ✌✟✎✔

① ✟ ✠✡✠✠✠✏☞ ✌✟✎☛

① ✟ ✠✡✠✠✎✑☞ ✌✟✎✕

① ✟ ✠✡✠✠✍✠☞ ✌✟✎✑

① ✟ ✠✡✠✠✑✍☞ ✌✟✎✍

① ✟ ✠✡✠✠☛☞ ✌✟✎✎

① ✟ ✠✡✠✠✏☞ ✌✟✎✠

① ✟ ✠✡✠✎✑☞ ✌✟✒

① ✟ ✠✡✠✍☞ ✌✟✏

① ✟ ✠✡✠✑✍☞ ✌✟✓

① ✟ ✠✡✠☛☞ ✌✟✔

① ✟ ✠✡✠✏☞ ✌✟☛

① ✟ ✠✡✎✑☞ ✌✟✕

① ✟ ✠✡✎✏☞ ✌✟✑

① ✟ ✠✡✍☛☞ ✌✟✍

① ✟ ✠✡✕✠☞ ✌✟✎

① ✟ ✠✡✔☛☞ ✌✟✠

◗

✷

✴ ✖✗✘

✷

s

r

✙✚

✛

✭

✜

✢✣

✤

✮

✜

✥

✐

✰

✦✧★✩ ✪ ✫✬ ✯

✱

♣

❋✲✳✯✵ ✶✸✹✺✯✻

✦✧★✩✼✽❋✾✿❀

❁❂

❃❄

❁❂

❃❅

❁❂

❃❆

❁

❁❂

❁❂

❅

❁❂

❄

❁❂

❇

❁❂

❈

❁❂

❉

❁❂

❊

❁

❁❂

❁❂

❅

❁❂

❄

❁❂

❇

❁❂

❈

Figure 2.3. HERA combined neutral current (NC) e

+

p reduced cross section and fixed-target

measurements as a function of momentum transfer, Q2, at fixed values of Bjorken-x. The error bars

indicate the total experimental uncertainty. A PDF set, called HERAPDF1.0 [50], is superimposed.

The bands represent the total uncertainty of the PDF fit. Dashed lines are shown for Q2 values not

included in the QCD analysis. (See [50] for further details.)

DGLAP (Dokshitzer, Gribov, Lipatov, Altarelli and Parisi) formalism [51–53] models these inter-

actions through splitting functions, and uses them to evolve perturbatively the renormalised parton

densities that contain the Q2 dependence. The implication of PDF evolution is that measuring par-

ton distributions for one scale, µ0, allows their prediction for any other scale, µ1, as long as both µ0

and µ1 are large enough for both αs(µ0) and αs(µ1) to be small.

The DGLAP formalism gives information on the evolution of the PDFs, though not on their

shape. The latter is derived using a combination of experimental data on the structure functions at

a given scale, Q2 = Q2

0

, and an initial analytical form. Seven functions should be determined, one

for the gluon and the others for each one of the light quarks and anti-quarks. Typically, specific

functional forms are postulated for the PDFs with a set of free parameters. The functional form

assumed for several PDF sets (such as CTEQ [8]), motivated by counting rules [54] and Regge

theory [55], is

fi(x,Q2

0) = x

αi(1−x)

βi gi(x) ,

(2.10)

18

2.2. Quantum Chromodynamics

where αi and βi are fit parameters and gi(x) is a function that asymptotically (x → 0 , x → 1) tends

to a constant.

QCD Factorisation

One of the reasons for the success of QCD as a predictive theory, is that the short-distance compo-

nent of the scattering process described by pQCD can be separated from the non-perturbative long-

distance component; this result is known as the factorisation theorem [48]. Factorisation implies

that perturbation theory can be used to calculate the hard scattering cross section, while universal

functions such as the PDFs1 can be included a posteriori to obtain the full theoretical prediction.

This takes the form,

dσfull (pA, pB,Q

2) =

∑

ab ∫ dxadxb fa/A (xa,µ2

F) fb/B (xb,µ2

F)×dσab→cd (αs (µ2

R),Q

2/µ2

R) , (2.11)

where the full and hard scattering cross sections are denoted by σfull and σab→cd respectively. Two

partons, a and b, respectively originating from hadrons A and B with momenta pA and pB, are the

constituents of the hard process. The integral is performed over the respective parton momenta

fractions xa and xb, weighted by fa/A and fb/B, which denote the parton momentum densities for the

two interacting partons. The sum is over parton flavours in the hadrons.

Factorisation is a byproduct of a procedure that absorbs singularities into physical quantities in

the same fashion as renormalisation. A new scale, µ2

F

, called the factorisation scale, is therefore

introduced in addition to the renormalisation scale µ2

R and the momentum transfer, Q2. Both µ2

R and

µ2

F

are generally chosen to be of the order of Q2. When truncating calculations at a given order, the

uncertainties due to the choice of scale need to be calculated in order to account for higher order

terms. The primary sources of uncertainty on QCD calculations are discussed next.

Uncertainties on QCD calculations

There are three main sources of uncertainties in the calculation of pQCD observables, summarized

in the following:

- higher order terms and scale dependence - the lack of knowledge of higher order

terms, neglected in the calculation, is estimated by varying the renormalization scale, µR,

usually by a factor of two with respect to the default choice. The factorization scale, µF, is

independently varied in order to evaluate the sensitivity to the choice of scale, where the PDF

evolution is separated from the partonic cross section. The envelope of the variation that these

changes introduce in an observable is taken as a systematic uncertainty;

- knowledge of the parameters of the theory - uncertainties on parameters of QCD, such

as the coupling constant, αs, and the masses of heavy quarks, are propagated into a measured

observable;

- PDF uncertainties - PDF uncertainties on an observable are evaluated differently for dif-

ferent PDF sets. They account for several factors; uncertainties on the data used to evaluate the

PDFs; tension between input data sets; parametrisation uncertainties; and various theoretical

uncertainties.

1 It has been shown experimentally that due to factorisation, PDFs are universal [56]; that is, they can be derived from

different physics processes and then used to provide full theoretical predictions independently from the calculation

of the hard scattering cross section.

19

2. Theoretical background

For the LHC, a major source of uncertainty is linked to the fact that pp interactions at

center-of-mass energies √s = 7 TeV, probe very low momenta of the partons in the proton, as

illustrated in figure 2.4. This low-x region is dominated by the gluon density [57], which is less well

constrained by measurements than the quark densities.

Example are shown in figure 2.5, where the PDFs being compared differ in the data samples

from which they were derived, and in the assumed value of αS. In figures 2.5a - 2.5b the fractional

uncertainties of the luminosities of quarks and gluons may be compared for several PDF sets. The

uncertainty on the gluon luminosities is larger than on those of the quarks, especially for low and

high values of x. An alternative way to illustrate this point is presented in figures 2.5c - 2.5d, where

the effect of the uncertainties on observable cross section predictions may be deduced; the gg cross

section for Higgs production is compared to that of Z production, the latter originating from q¯q

interactions. The relative spread of cross section perdictions depending on the gluonic PDFs is

roughly twice that of the quarks’.

HERA measurements at low-x, and in particular forward jet production [59, 60], indicate that the

DGLAP dynamic used in deriving the PDFs may not be sufficient to describe the interactions in this

region [61]. This may indicative the need to include higher order pQCD corrections, or use of the

formalisim of Balitsky, Fadin, Kuraev and Lipatov (BFKL) [62–64].

2.3. Full description of a hard pp collision

A full description of the final state of a pp collision incorporates two elements. The first is the hard

scattering, involving a large transfer of transverse momentum and calculable in pQCD. The second

part pertains to non perturbative effects, taking into account low-pt interactions and hadronization.

The two aspects of the computation are discussed in the following sections.

2.3.1. The hard scattering

The hard scattering is computed at a fixed order (in the strong coupling constant) in perturbation

theory. The dominant contributions to jet cross sections arise from Feynman diagrams that con-

tribute to jet production at leading order (LO), known as (2→2) diagrams. Some examples of LO

diagrams are shown for quark/quark t-channel scattering, quark/anti-quark s-channel annihilation,

and gluon/gluon t-channel scattering in figures 2.6a - 2.6c. A calculation at next-to-leading order

(NLO) may include either a (2→2) process with one virtual loop, as shown in figure 2.6d, or a

(2→3) interaction, as in figure 2.6e, where one of the incoming or outgoing partons radiates a third

parton.

The results of a full NLO calculation of the differential dijet mass cross section, using the NLO-

JET++ package [65], are compared to ATLAS data in chapter 8.

2.3.2. Non-perturbative effects

As mentioned above, complete pQCD calculations are performed up to a fixed order in αs. However,

the enhanced soft-gluon radiation and collinear configurations at higher orders can not be neglected.

These are taken into account in the parton shower (PS) approximation, that sums the leading con-

tributions of such topologies to all orders. MC generator programs include the PS approximation,

20

2.3. Full description of a hard pp collision

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

0

10

0

10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

fixed

target

HERA

x

1,2

= (M/7 TeV) exp(±y)

Q = M

7 TeV LHC parton kinematics

M = 10 GeV

M = 100 GeV

M = 1 TeV

M = 7 TeV

6

6

y =

4

0

2

2

4

Q

2

(G

e

V

2

)

x

WJS2010

Figure 2.4. The (x,Q2) kinematic plane for LHC at √s = 7 TeV, where x denotes the fraction

of the momentum of the proton carried by an interacting parton, and Q2 stands for the momentum

transfer of the interaction. Shown also are the ranges covered by HERA and fixed-target experi-

ments. The x and Q2 ranges probed by the production of mass, M, or a hard final state at fixed

rapidity, y, are indicated by the dashed lines. (Figure taken from http://projects.hepforge.

org/mstwpdf/, courtesy of J. Stirling.)

21

2. Theoretical background

/ ss

-3

10

-2

10

-1

10

Fractional uncertainty (68% C.L.) -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

= 7 TeV)

s

) luminosity at LHC (

q(q

q

Σ

W Z

MSTW08 NLO

CTEQ6.6

CT10

NNPDF2.1

G. Watt (March 2011)

/ ss

-3

10

-2

10

-1

10

Fractional uncertainty (68% C.L.) -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

(a)

/ ss

-3

10

-2

10

-1

10

Fractional uncertainty (68% C.L.) -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

= 7 TeV)

s

gg luminosity at LHC (

tt

120

180 240

(GeV)

H

M

MSTW08 NLO

CTEQ6.6

CT10

NNPDF2.1

G. Watt (March 2011)

/ ss

-3

10

-2

10

-1

10

Fractional uncertainty (68% C.L.) -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

(b)

)2

Z

(M

S

α

0.114

0.116

0.118

0.12

0.122

0.124

(pb)

Hσ

10

10.5

11

11.5

12

12.5

13

68% C.L. PDF

MSTW08

CTEQ6.6

CT10

NNPDF2.1

HERAPDF1.0

ABKM09

GJR08

= 120 GeV

H

= 7 TeV) for M

s

H at the LHC (

→

NLO gg

S

α

Outer: PDF+

Inner: PDF only

Vertical error bars

G. Watt (April 2011)

)2

Z

(M

S

α

0.114

0.116

0.118

0.12

0.122

0.124

(pb)

Hσ

10

10.5

11

11.5

12

12.5

13

(c)

)2

Z

(M

S

α

0.114

0.116

0.118

0.12

0.122

0.124

) (nb)- l+ l

→

0

B(Z⋅

0 Zσ

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

68% C.L. PDF

MSTW08

CTEQ6.6

CT10

NNPDF2.1

HERAPDF1.0

ABKM09

GJR08

= 7 TeV)

s

at the LHC (

-l+

l

→

0

NLO Z

S

α

Outer: PDF+

Inner: PDF only

Vertical error bars

G. Watt (April 2011)

)2

Z

(M

S

α

0.114

0.116

0.118

0.12

0.122

0.124

) (nb)- l+ l

→

0

B(Z⋅

0 Zσ

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

(d)

Figure 2.5. (a)-(b) Fractional uncertainties on the NLO parton-parton luminosities for quarks

and for gluons, derived using different PDF sets, as indicated in the figures. The uncertainties are

shown as a function of the fractional center-of-mass energy of the partonic system responsible for

the scattering, denoted by √ˆs/s.

(c)-(d) Cross sections for Higgs and Z0 production at √s = 7 TeV for different PDF sets, as indicated

in the figures. The cross sections are shown as a function of the strong coupling constant (at the

scale of the Z), αS(M2

Z), which is used in the derivation of the PDFs. The error bars represent

the uncertainties on the respective PDFs excluding (inner) or including (outer) the uncertainty on

αS(M2

Z). The dashed lines interpolate the cross section predictions calculated with each PDF set for

different values of αS(M2

Z).

(Figures are taken from [58].)

22

2.3. Full description of a hard pp collision

(a)

(b)

(c)

(d)

(e)

Figure 2.6. Examples for leading-order ((a)-(c)) and next-to-leading order ((d)-(e)) Feynman dia-

grams for jet production in proton-proton collisions at the LHC; t-channel scattering (a), quark/anti-

quark s-channel annihilation (b), gluon/gluon t-channel scattering (c), a (2→2) diagram with a vir-

tual loop (d), a (2→3) diagram where the third outgoing parton is produced via real emission from

another parton (e).

as well as models to reproduce non-perturbative effects, such as hadronization and the underlying

event.

Since non-perturbative physics models are by necessity deeply phenomenological, they usually

account for the majority of parameters incorporated into event generators. For instance, typical

hadronization models require parameters to describe e.g., the kinematic distribution of transverse

momentum in hadron fragmentation, baryon-to-meson ratios, strangeness and suppression of η

and η′ mesons, and the assignment of orbital angular momentum to final state particles. Event

generators are therefore tuned to data in specific regions of phase-space. At times it is not possible

to match all features of the data, e.g., simultaneous description of both the transverse momentum and

the multiplicity distributions of charged particles in hadron-hadron collisions [66]. Different tunes

are usually compared in order to asses the theoretical uncertainty associated with non-perturbative

processes.

The different elements of event generators, apart from the hard scattering itself, are discussed

next.

Parton showers

The PS approximation describes successive parton emission from the partons taking part in the hard

interaction. In principle, the showers represent higher-order corrections to the hard subprocess.

However, it is not feasible to calculate these corrections exactly. Instead, an approximation scheme

is used, in which the dominant contributions are included in each order. These dominant contribu-

tions are associated with collinear parton splitting or soft gluon emission, illustrated in figure 2.7.

The evolution of the shower is usually governed by the DGLAP equations, though at low-x,

as mentioned above, BFKL dynamics may be required. Numerical implementation of the parton

23

2. Theoretical background

(a)

(b)

(c)

Figure 2.7. Examples of leading-order (1→2) Feynman diagrams for the splitting of quarks and

gluons; a quark emitting a gluon (a), a gluon emitting another gluon (b), a gluon splitting into a

quark/anti-quark pair (c).

shower is achieved using the Sudakov form factors [67]; these represent the probability that a parton

does not branch between some initial scale and another, lower scale. In each step, as a branching

a → bc occurs from scale ta, subsequent branchings are derived from the scales tb and tc of the prod-

ucts of the initial state. Branchings can be angle- or transverse momentum-ordered. For the former,

the opening angles between each successive branchings become smaller; for the latter, emissions

are produced in decreasing order of intrinsic pt. Successive branching stops at a cutoff scale of the

order of ΛQCD, after producing a high-multiplicity partonic state.

Since quarks and gluons can not exist isolated, MC programs contain models for the hadroniza-

tion of the partons into colorless hadrons, discussed in the following.

Hadronization

The hypothesis of local parton-hadron duality states that the momentum and quantum numbers of

hadrons follow those of their constituent partons [68]. This makes up the general guideline of all

hadronization models. There exist two main models of hadron production in the popular physics

generators, the string model and the cluster model.

The string model [69] for hadronization, depicted schematically in figure 2.8a, is based on

an observation from lattice simulations of QCD; at large distances, the potential energy of colour

sources, such as heavy quark/anti-quark (q¯q) pairs, increases linearly with their separation. This

indicates a distance-independent force of attraction, thought to be due to the self-attraction of the

gluonic field.

In the model, the field between each q¯q pair is represented by a string with uniform energy per

unit length. As the q and the ¯q move apart from each other, the energy of the color field increases

and the string connecting the two is tightened. Eventually the string breaks, and its two ends form

a new quark/anti-quark pair. If the invariant mass of either of these string pieces is large enough,

further breaks may occur in addition. The string break-up process is assumed to proceed until only

on-mass-shell hadrons remain. In the simplest approach of baryon production, a diquark (D) is

treated just like an ordinary anti-quark; a string can break either into a quark/anti-quark or into a

diquark/anti-diquark pair, leading to three-quark states.

The cluster model [70] for hadronization is based on the so-called preconfinement property

of QCD, discovered by Amati and Veneziano [71]. They showed that at evolution scales, q, much

smaller than the scale of the hard subprocess, the partons in a shower are clustered in colourless

groups. These groups have an invariant mass distribution that is independent of the nature and

24

2.3. Full description of a hard pp collision

scale of the hard subprocess, depending only on q and on ΛQCD. It is then natural to identify these

clusters at the hadronization scale, Q0, as proto-hadrons that later decay into the observed final-state

hadrons.

In practical terms, at a scale around Q0, gluons from the PS are split into light quark/anti-quark

or diquark/anti-diquark pairs, as illustrated in figure 2.8b. Color-singlet clusters are formed from

the different pair combinations; mesonic (q¯q and D ¯D), barionic (qD) and anti-barionic ( ¯q ¯D). The

clusters thus formed are fragmented into two hadrons. If a cluster is too light to decay into two

hadrons, it is taken to represent the lightest single hadron of its flavor; its mass is therefore shifted

to the appropriate value by an exchange of momenta with a neighboring cluster. If the cluster is too

heavy, it decays into two clusters, which are further fragmented into hadrons.

(a)

(b)

Figure 2.8. Schematic illustration of hadronization in the string (a) and in the cluster (b) models.

The underlying event

In events that contain a hard subprocess, there is extra hadron production that cannot be ascribed

to showering from the coloured partons participating in the subprocess, known as the underlying

event (UE). Furthermore, this extra activity is greater than that in so-called minimum-bias events

(collisions that do not yield an identifiable hard subprocess). The UE is believed to arise from

collisions between those partons in the incoming hadrons that do not directly participate in the hard

subprocess.

The most common hard subprocess at the LHC is elastic gluon-gluon scattering, gg → gg. The

leading-order differential cross section for this subprocess diverges at zero momentum transfer, due

to the exchange of a massless virtual gluon. This divergence is presumably regularized below some

momentum transfer, tmin, by higher-order and non-perturbative effects. Nevertheless, for reasonable

values of tmin, the integrated cross section for gluon-gluon scattering is very large, larger even than

the total proton-proton (pp) scattering cross section. This result indicates that the average number

of gluon-gluon collision per pp collisions is greater than one. Including the cross sections for elastic

scattering of quarks, anti-quarks and gluons in all possible combinations (all of which diverge in

leading order), multiple parton interactions (MPI) are found to be highly probable

This is the basis on which modern event generators model both minimum-bias collisions and

the UE. To account for the extra hadron production when a hard subprocess is present, an event

25

2. Theoretical background

generator must model the impact parameter structure of hadron-hadron collisions. The partons in

each incoming hadron are distributed over a transverse area of the order of 1 fm2. The impact

parameter of a collision is the transverse distance between the centroids of these areas before the

collision. When the impact parameter is large, the areas overlap little and the collision is peripheral;

this configuration is associated with a low probability of a hard parton-parton interaction and few

MPI. On the other hand, at small impact parameter values, the collision is central and has a large

overlap of areas; several multiple interactions and a higher probability of a hard interaction are

therefore expected.

To summarize, the presence of a hard subprocess is correlated with more MPI and a higher

level of UE activity. Most of the multiple-interactions are soft, though hard MPI is also possible.

As mentioned in chapter 1, hard MPI are an important background for e.g., new physics signals

involving milti-jet final states. As such, hard MPI are given special attention beyond the general

discussion in the context of the UE. The most simple (and often most prominent) case, that of

double parton scattering, is reviewed in the next section.

2.4. Double parton scattering

The formalism to deal with (semi-)hard double parton scattering in hadronic interactions at

center-of-mass energy √s [21, 22] may be summarized by

σDPS

(A,B)(s) =

m

2

∑

i, j,k,l ∫ Γij(x1,x2,d;QA,QB) ˆσ(A)

ik

(x1,x′1,s) Γkl(x′1,x′2,d;QA,QB)

× ˆσ(B)

jl

(x2,x′2,s) dx1 dx2 dx′1 dx′2 d2d ,

(2.12)

where σDPS

(A,B)

is the differential double parton scattering cross section for the inclusive production of

a combined system A+B at a given √s; the terms ˆσ(A)

ik

denote the differential partonic cross sections

for the production of a system A in the collision of partons i and k; the terms Γij(x1,x2,d;QA,QB)

represent double parton distribution functions (DPDFs); and the parameter m is a symmetry factor

such that m = 1 if A = B and m = 2 otherwise. The integration over the momentum fractions x1

and x2 is constrained by energy conservation, such that (x1 +x2 ≤ 1). Summation over all possible

parton combinations is performed. A sketch of double parton scattering is shown in figure 2.9 for

illustration.

The DPDFs, Γij(x1,x2,d;QA,QB), may be loosely interpreted as the inclusive probability distri-

bution to find a parton i (j) with longitudinal momentum fraction x1 (x2) at scale QA (QB) in the

proton, with the two partons separated by a transverse distance d. The scale QA (QB) is given by

the characteristic scale of subprocess A (B). It is assumed that the DPDFs may be decomposed into

longitudinal and transverse components,

Γij(x1,x2,d;QA,QB) ≃ Dij(x1,x2;QA,QB) F(d) .

(2.13)

The longitudinal component, Dij(x1,x2;QA,QB), has a rigorous interpretation in leading order

pQCD, as the inclusive probability of finding a parton i with momentum fraction x1 at scale QA,

as well as and a parton j with momentum fraction x2 at scale QB in a proton. Accurate predic-

tion of double parton scattering cross sections and of event signatures requires good modelling of

Dij(x1,x2;QA,QB) and of the transverse component, F(d). In particular, one must correctly take

26

2.4. Double parton scattering

p

p

b

a

c

d

i

j

k

l

Figure 2.9. Sketch of a double parton scattering process, in which the active partons originating

from one proton are i and j and from the other are k and l. The two hard scattering subprocess are

A(i k → a b) and B(j l → c d).

account of the effects of correlations in both longitudinal momenta and transverse positions in these

functions.

Correlations between the partons in transverse space are highly significant; at the very least, they

must tie the two partons together within the same hadron. However, their precise calculation is not

possible using perturbation theory. Existing models typically use Gaussian or exponential forms (or

their combination) to describe F(d) [36,72]. The transverse component is usually expressed simply

as

σeff(s) = [∫ d2d(F(d))2]

−1

.

(2.14)

The quantity σeff(s) is defined at the parton-level, and has the units of a cross section. In the

formalism outlined here, it is independent of the process and of the phase-space under consideration.

Naively, it may be related to the geometrical size of the proton. That is, given that one hard scattering

occurs, the probability of the other hard scattering is proportional to the flux of accompanying

partons; these are confined to the colliding protons, and therefore their flux should be inversely

proportional to the area (cross section) of a proton. This leads to an estimate of σeff ≈ πR2

p ≈ 50 mb,

where Rp is the proton radius. Alternatively, σeff may be connected to the inelastic cross section,

which would lead to σeff ≈ σinel ≈ 70 mb at

√s = 7 TeV [73, 74]. For hard interactions, assuming

uncorrelated scatterings, σeff can be estimated from the gluon form factor of the proton [75] and

comes out to be ∼ 30 mb.

A number of measurements of σeff have been performed in pp or p ¯p collisions at different center-

of-mass energies, as specified in table 2.1. The energy dependence of the measured values of σeff

yielded an increase from about 5 mb at the lowest energy (63 GeV) to about 15 mb at LHC ener-

gies (7 TeV). Attempts to explain the differences between these values have used the Constituent

Quark Model [76, 77], such as in [78], or have introduced non-trivial correlations between the two

scattering systems, as in [16–20]. A complete explanation, however, is still elusive. A recent argu-

ment, suggested in [79–81] may resolve the discrepancy; it goes as follows. In DPS, two partons

from one proton collide with two partons from the other proton. The two partons from a given

proton can originate from the non-perturbative hadron wave function or, alternatively, emerge from

perturbative splitting of a single parton from the hadron. The former represents a double-(2→2)

interaction, and the latter a (3→4) interaction. The (3→4) process could explain the difference

27

2. Theoretical background

Experiment

√s [GeV]

Final state

σeff [mb]

[23]

AFS (pp), 1986

63

4 jets

∼5

[24]

UA2 (p ¯p), 1991

630

4 jets

>8.3 (95% C.L)

[25]

CDF (p ¯p), 1993

1800

4 jets

12.1

+10.7

−5.4

[26]

CDF (p ¯p), 1997

1800

γ + 3-jets

14.5 ± 1.7

+ 1.7

−2.3

[27]

DØ (p ¯p), 2010

1960

γ + 3-jets

16.4 ± 0.3 ± 2.3

[28] ATLAS (pp), 2012

7000

W + 2-jets

15 ± 3

+ 5

−3

Table 2.1. Summary of published measurements of σeff.

between the experimental value of σeff of ∼ 15 mb, and the expected value of ∼ 30 mb.

In the absence of a rigorous formalism, measurements of σeff typically assume a simple factor-

ization ansatz for the DPDFs,

Dij(x1,x2;QA,QB) ≃ Di(x1;QA)Dj(x2;QB) .

(2.15)

The differential double parton scattering cross section defined in equation (2.12) therefore reduces

to

σDPS

(A,B) =

m

2

σAσB

σeff

.

(2.16)

The assumption of factorization is problematic. For one, this naive representation does not obey

the relevant momentum and number sum rules. In addition, while previous experiments suggest

that approximate factorisation holds at moderately low x [26], this can not be true for all values

of x. Namely, if Di(x1;QA) and Dj(x2;QB) each satisfy DGLAP evolution, then the naive product,

Di(x1;QA)Dj(x2;QB), can not be a solution of the double-DGLAP equations (dDGLAP), suggested

e.g., in [17]. In order to use the factorization ansatz, the unknown correlations are absorbed into

σeff, which then possibly becomes dependant on the process (and respective phase-space) under

consideration.

A measurement of the rate of double parton scattering in four jet events in ATLAS is presented

in this thesis, using the simplified form of the effective cross section given in equation (2.15). Only

the double-(2→2) topology is considered in the analysis, as discussed in chapter 9.

28

3. The ATLAS experiment at the LHC

3.1. The Large Hadron Collider

The Large Hadron Collider (LHC) is the world’s largest and highest-energy particle accelerator.

LHC is a proton-proton (pp) collider, located at the Franco-Swiss border near Geneva, Switzerland.

It lies in a tunnel 27 km in circumference at an average depth of 100 meters. The tunnel houses

1232 superconducting bending dipole magnets, cooled using liquid helium to an operating tempera-

ture of 1.9 K, producing a magnetic field of about 8 T. The use of dipole magnets allows to keep pro-

tons traveling clockwise and counter-clockwise on orbit at the same time. In total, 392 quadrupole

magnets are used to keep the beams focused and to collide them at the four interaction points (IPs)

of the LHC experiments. The design center-of-mass energy of the LHC is √s = 14 TeV.

Protons are produced by ionizing hydrogen atoms in an electric field. They are injected into

RF cavities and accelerated to 750 keV. The beam is then transmitted to the LINAC 2, a linear

accelerator, which increases the energy to 50 MeV. The protons are accelerated to 1.4 GeV by the

Proton Synchrotron Booster and then further to 26 GeV by the Proton Synchrotron. Next, the Super

Proton Synchrotron accelerates the protons to 450 GeV, the minimum energy required to maintain

a stable beam in the LHC. Finally, the LHC accelerates them to the operating energy.

During the first two years of operation (2010-2011), the LHC operated at a center-of-mass energy

of √s= 7 TeV, with 3.5 TeV per proton. This center-of-mass energy has been chosen to ensure a safe

operating margin for the magnets in the accelerator. So far, up to the end of the 2011 run, the LHC

delivered 5.61 fb−1

total integrated luminosity with a peak luminosity of 3.65×1033 cm−2s−1 [82].

The bunch separation was 50 ns for most of the running period. The full 2010 and most of the 2011

datasets are used in this analysis.

3.2. The ATLAS detector

ATLAS (A Toroidal Lhc ApparatuS) [83, 84] is a general-purpose detector surrounding IP 1 of

the LHC. ATLAS consists of three main sub-systems, the inner detector (ID), the calorimeters and

the muon spectrometer (MS). Figure 3.1a shows a schematic view of the ATLAS detector and its

sub-systems.

Detector sub-systems

The ID is used to measure the tracks of charged particles. It covers the pseudo-rapidity range,

|η| < 2.5 1, and has full coverage in azimuth. It is made of three main components, arranged in con-

centric layers, all of which are immersed in a 2 T field provided by the inner solenoid magnet. Three

1 The coordinate system used by ATLAS is a right-handed Cartesian coordinate system. The positive z-direction is

defined as the direction of the anti-clockwise beam. Pseudo-rapidity is defined as η = lntan(θ/2), where θ is the

angle with respect to the z-axis. The azimuthal angle in the transverse plane φ is defined to be zero along the x-axis,

which points toward the center of the LHC ring.

29

3. The ATLAS experiment at the LHC

layers of silicon pixel detectors provide a two-dimensional hit position very close to the interaction

point. Silicon microstrip detectors are then used in the next four layers, providing excellent position

resolution for charged tracks. A transition-radiation detector is the final component of the tracker,

with poorer position resolution with respect to the silicon, but providing many measurement points

and a large lever-arm for track reconstruction in addition to particle identification capabilities.

The ATLAS calorimeter, shown schematically in figure 3.1b, is the principal tool used in the

analysis. The calorimeter is composed of several sub-detectors. The liquid argon (LAr) electro-

magnetic (EM) calorimeter is divided into one barrel (|η| < 1.475) and two end-cap components

(1.375 < |η| < 3.2). It uses an accordion geometry to ensure fast and uniform response and fine

segmentation for optimum reconstruction and identification of electrons and photons. The hadronic

scintillator tile calorimeter consists of a barrel covering the region, |η| < 1.0, and two extended

barrels in the range 0.8 < |η| < 1.7. The LAr hadronic end-cap calorimeter (1.5 < |η| < 3.2) is

located behind the end-cap electromagnetic calorimeter. The forward calorimeter covers the range

3.2 < |η| < 4.9 and also uses LAr as the active material.

The MS forms the outer part of the ATLAS detector and is designed to detect muons exiting

the barrel and end-cap calorimeters and to measure their momentum in the pseudo-rapidity range

|η| < 2.7. It is also designed to trigger on muons in the region |η| < 2.4. The MS operates inside an

air-core toroid magnet system with a peak field in the coil windings of 4 T. The precision momentum

measurement is performed by the Monitored Drift Tube chambers covering the pseudo-rapidity

range |η| < 2.7 (except in the innermost end-cap layer where their coverage is limited to |η| < 2.0).

In the forward region (2.0 < |η| < 2.7), Cathode-Strip Chambers are used in the innermost tracking

layer. The capability to trigger on muon tracks is achieved by a system of fast trigger chambers

capable of delivering track information within a few tens of nanoseconds after the passage of the

particle. In the barrel region (|η| < 1.05), Resistive Plate Chambers were selected for this purpose,

while in the end-cap (1.05 < |η| < 2.4) Thin Gap Chambers were chosen.

The Trigger System

The ATLAS detector has a three-level trigger system consisting of Level 1 (L1), Level 2 (L2) and

Event Filter (EF). The L1 trigger rate at design luminosity is approximately 75 kHz. The L2 and

EF triggers reduce the event rate to approximately 200 Hz. Another trigger system used in ATLAS

relies on the minimum-bias trigger scintillators (MBTS). The MBTS trigger requires one hit above

threshold from either one of the sides of the detector. The different triggers used in the analysis are

described in further detail in chapter 6, section 6.2.

Luminosity Measurement

Accurate measurement of the delivered luminosity is a key component of the ATLAS physics pro-

gram. For cross section measurements of SM processes, the uncertainty on the delivered luminosity

is often one of the dominant systematic uncertainties.

The instantaneous luminosity of proton-proton collisions can be calculated as

L =

Rinel

σinel

,

(3.1)

30

3.2. The ATLAS detector

(a)

(b)

Figure 3.1. Schematic view of the ATLAS detector (a) and of the ATLAS calorimeter system (b).

31

3. The ATLAS experiment at the LHC

where Rinel is the rate of pp interactions and σinel is the inelastic cross section. Any detector sensitive

to inelastic pp interactions can be used as a source for relative luminosity measurement. However,

these detectors must be calibrated using an absolute measurement of the luminosity.

The recorded luminosity can be written as

L =

µvisnb frev

σvis

,

(3.2)

where µvis is the visible number of interactions per bunch crossing in the detector, nb is the number

of bunch pairs colliding in ATLAS, frev = 11245.5 Hz is the LHC revolution frequency and σvis is

the visible cross section, to be determined via calibration for each detector.

The calibration is done using dedicated beam separation scans, also known as van der Meer

scans [85], where the two beams are stepped through each other in the horizontal and vertical planes

to measure their overlap function. The delivered luminosity is measured using beam parameters,

L =

nb frevn1n2

2πΣxΣy

,

(3.3)

where n1 and n2 are respectively the bunch populations (protons per bunch) in beam 1 and in beam 2

(together forming the bunch charge product), and Σx and Σy respectively characterize the horizontal

and vertical profiles of the colliding beams.

By comparing the delivered luminosity to the peak interaction rate, µMax

vis

, observed by a given

detector during the van der Meer scan, it is possible to determine the visible cross section,

σvis = µMax

vis

2πΣxΣy

n1n2

.

(3.4)

Two detectors are used to make bunch-by-bunch luminosity measurements in 2010 and in 2011,

LUCID and BCM. LUCID is a Cerenkov detector specifically designed for measuring the luminos-

ity in ATLAS. It is made up of sixteen mechanically polished aluminum tubes filled with C4F10 gas

surrounding the beampipe on each side of the IP at a distance of 17 m, covering the pseudo-rapidity

range, 5.6 < |η|< 6.0. The beam conditions monitor (BCM) consists of four small diamond sensors

on each side of the ATLAS IP arranged around the beam-pipe in a cross pattern. The BCM is a fast

device, primarily designed to monitor background levels. It is also capable of issuing a beam-abort

request in cases of possible damage to ATLAS detectors, due to beam losses.

The relative uncertainty on the luminosity scale for pp collisions during 2010 [86] and 2011 [85]

was found to be, respectively ±3.4 and ±3.9%2.

2 The uncertainty on the luminosity in 2011 quoted here is larger than the value given in [85]. The increased value is

a consequence of recent studies by the Luminosity group in ATLAS, which show significant non-linear correlations

between the van der Meer scans in the horizontal and vertical directions [87]. These correlations were not previously

known and are not yet fully understood.

32

4. Monte Carlo simulation

Any analysis involving a complex detector such as ATLAS, requires a detailed detector Monte Carlo

(MC) simulation. The MC is used in order to compare distributions of observables, as simulated by

physics generators, with the data. In addition, it is used to study the performance of the detector by

estimating reconstruction efficiencies, geometrical coverage, the performance of triggers etc.

A description of the MC generators and of the ATLAS detector simulation used in the analysis is

presented in the following.

4.1. Event generators

The four-momenta of particles produced in pp collisions at the LHC are simulated using various

event generators. An overview of MC event generators for LHC physics can be found in [88]. The

different MC samples used for comparison with data taken during 2010 and during 2011 are de-

noted respectively as MC10 and MC11. The following event generators, using different theoretical

models, are utilized in the analysis.

Pythia

PYTHIA [89] simulates non-diffractive pp collisions using a 2 → 2 matrix element at leading order

in the strong coupling to model the hard subprocess. It uses pt-ordered parton showers to model

additional radiation in the leading-logarithmic approximation [90]. Multiple parton interactions [36,

91], as well as fragmentation and hadronization, based on the Lund string model [69], are also

simulated.

Several PYTHIA 6 tunes utilizing different parton distribution function (PDF) sets are used. The

nominal version used in this analysis employs the modified leading-order PDF set, MRST LO* [7].

The parameters used for tuning the underlying event include charged particle spectra measured by

ATLAS in minimum bias collisions [92]. The samples used to compare with the 2010 and with the

2011 data respectively use the AMBT1 [93] and AMBT2B [94] tunes.

Several additional combinations of PYTHIA versions, PDF sets and underlying event tunes are

used in chapter 8 in order to evaluate non-perturbative corrections to the dijet mass cross section.

These include the following; PYTHIA 6.425 with the MRST LO* PDF set and the AUET2B [94]

tune; PYTHIA 6.425 with the CTEQ6L1 [8] PDF set and the AMBT2B and AUET2B tunes;

PYTHIA 8 (v150) [95] with the MRST LO** PDF set and 4C [96] tune.

Herwig, Herwig++

HERWIG uses a leading order (2→2) matrix element supplemented with angular-ordered parton

showers in the leading-logarithm approximation [97–99]. The so called cluster model [70] is

used for the hadronization. Multiple parton interactions are modelled using JIMMY [100]. The

model parameters of HERWIG+JIMMY have been tuned to ATLAS data (AUET1 tune) [101]. The

MRST LO* PDF set is used.

33

4. Monte Carlo simulation

HERWIG++ [102] (v2.5.1) is based on HERWIG, but redesigned in the C++ programming lan-

guage. The generator contains a few modelling improvements. It also uses angular-ordered parton

showers, but with an updated evolution variable and a better phase-space treatment. Hadroniza-

tion is performed using the cluster model, as in HERWIG. The underlying event and soft inclusive

interactions are described using a hard and soft multiple partonic interactions model [72]. The

MRST LO* and the CTEQ6L1 PDF sets are used with the UE7000-3 [94] underlying event tune, in

order to evaluate non-perturbative corrections to the dijet mass cross section in chapter 8.

Alpgen

ALPGEN is a tree level matrix-element generator for hard multi-parton processes (2→n) in hadronic

collisions [103]. It is interfaced to HERWIG to produce parton showers in the leading-logarithmic

approximation. Parton showers are matched to the matrix element with the MLM matching

scheme [104]. For the hadronization, HERWIG is interfaced to JIMMY in order to model soft mul-

tiple parton interactions. The PDF set used is CTEQ6L1. ALPGEN is used in chapter 8 in order to

evaluate the systematic uncertainty associated with the unfolding procedure of the dijet mass cross

section measurement.

Sherpa

SHERPA is a general-purpose tool for the simulation of particle collisions at high-energy colliders,

using a tree-level matrix-element generator for the calculation of hard scattering processes [105].

The emission of additional QCD partons off the initial and final states is described through a

parton-shower model. To consistently combine multi-parton matrix elements with the QCD par-

ton cascades, the approach of Catani, Krauss, Kuhn and Webber [106] is employed. A simple

model of multiple interactions is used to account for underlying events in hadron–hadron collisions.

The fragmentation of partons into primary hadrons is described using a phenomenological cluster-

hadronization model [107]. A comprehensive library for simulating tau-lepton and hadron decays

is provided. Where available form-factor models and matrix elements are used, allowing for the

inclusion of spin correlations; effects of virtual and real QED corrections are included using the

approach of Yennie, Frautschi and Suura [108]. The CTEQ6L1 PDF sets together with the default

underlying event tune are used. The CKKW matching scale is set at 15 GeV, where the latter refers

to the energy scale in which matching of matrix elements to parton showers begins. The implication

of this choice is that partons with transverse momentum above 15 GeV in the final state, necessarily

originate from matrix elements, and not from the parton shower.

SHERPA is employed in the analysis as part of the double parton scattering measurement

in chapter 9. events are generated without multiple interactions, by setting the internal flag,

MI HANDLER=None. The generated events serve as input to a neural network.

4.2. Simulation of the ATLAS detector

Event samples produced using the different event generators described above are passed through

the full ATLAS detector simulation and are reconstructed as the data. The GEANT software

toolkit [109] within the ATLAS simulation framework [110] propagates the generated particles

through the ATLAS detector and simulates their interactions with the detector material. The energy

deposited by particles in the active detector material is converted into detector signals with the same

format as the ATLAS detector read-out. The simulated detector signals are in turn reconstructed

with the same reconstruction software as used for the data.

34

4.3. Bunch train structure and overlapping events

In GEANT the model for the interaction of hadrons with the detector material can be specified for

various particle types and for various energy ranges. For the simulation of hadronic interactions in

the detector, the GEANT set of processes called QGSP BERT [111] is chosen. In this set of processes,

the Quark Gluon String model [112–116] is used for the fragmentation of the nucleus, and the

Bertini cascade model [117–120] for the description of the interactions of hadrons in the nuclear

medium.

The GEANT simulation, and in particular the hadronic interaction model for pions and protons,

has been validated with test-beam measurements for the barrel [121–125] and endcap [126–128]

calorimeters. Agreement within a few percent is found between simulation and data for pion mo-

menta between 2 and 350 GeV. Further tests have been carried out in-situ comparing the single

hadron response, measured using isolated tracks and identified single particles [129, 130]. Good

agreement between simulation and data is found for particle momenta from a few hundred MeV to

6 GeV.

Studies of the material of the inner detector in front of the calorimeters have been performed us-

ing secondary hadronic interactions [131]. Additional information is obtained from studying photon

conversions [132] and the energy flow in minimum bias events [133]. The ATLAS detector geom-

etry used in the 2010 and in the 2011 simulation campaigns reflects the best current knowledge of

the detector at the time the simulations were made. Subsequently, compared to MC10, a more de-

tailed description of the geometry of the LAr calorimeter was used for MC11. The improvement in

understanding of the geometry introduced an increased calorimeter response to pions below 10 GeV

of about 2%.

The MC11 simulation is made up of different MC-periods. Each MC-period represents different

data-taking conditions, such as malfunctioning hardware or changes in the amount of in-time pile-

up, additional pp collisions, coinciding with the hard interaction. Events in the different MC-

periods are given relative weights according to the integrated luminosity of the respective data-

taking periods they represent.

A note regarding the simulation in chapter 7; before the start of data-taking in 2011, a transi-

tional MC simulation was created, denoted as MC10b. MC10b was generated as part of the MC10

campaign. It used the 2010 simulation settings as described above, with the exception of highly

increased pile-up conditions matching the expected characteristics of the 2011 data. The study

described in chapter 7 was originally performed using an MC10b PYTHIA sample. When MC11

became available, most of the results were reproduced; those which were not, are presented here us-

ing the previous version of the simulation. Unless otherwise indicated, MC11 is used in conjecture

with the 2011 data.

4.3. Bunch train structure and overlapping events

The LHC bunch train structure of the 2010 and the 2011 data is modelled in MC. In MC10 the

simulated collisions are organised in double trains with 225 ns separation. Each train contains eight

filled proton bunches with a bunch separation of 150 ns. In MC11 the simulation features four trains

with 36 bunches per train and 50 ns spacing between the bunches.

The MC samples are generated with additional minimum bias interactions, using PYTHIA 6 with

the AMBT1 underlying event tune and the MRST LO* PDF set for MC10, and PYTHIA 6 with the

AUET2B tune and the CTEQ6L1 PDF set for MC11. These minimum bias events are used to form

pile-up events, which are overlaid onto the hard scattering events. The number of overlaid events

35

4. Monte Carlo simulation

follows a Poisson distribution around the average number of additional pp collisions per bunch

crossing, as measured in the experiment. The average number of additional interactions depends

on the instantaneous luminosity (see equation (7.1) in section 7.1.1); it is thus greater in MC11

compared to MC10, following the trend in the data.

The small separation between bunches in the 2011 data (and respectively in MC11) requires

inclusion of the effect of out-of-time pile-up1 in the simulation. The properties of this effect depend

on the distance of the hard scatter events from the beginning of the bunch train. The first ten

(approximately) bunch crossings are characterized by increasing out-of-time pile-up contributions

from the previous collisions. For the remaining 26 bunch crossings in a train, the out-of-time pile-up

is stable within the bunch-to-bunch fluctuations in proton intensity at the LHC.

1 A feature of the 2011 data is that the signal in the calorimeter is sensitive to collisions from several consecutive

bunch crossings; out-of-time pile-up refers to this fact. See section 7.1 for a detailed discussion.

36

5. Jet reconstruction and calibration

5.1. Jet reconstruction algorithms

Jets originate from the fragmentation of partons. Due to color flow at the parton-level, there is no

one-to-one correspondence between the hadrons inside a jet and the partons which initiated the jet.

Because the measurements are performed at the hadron-level and the theoretical expectations at the

parton-level, a precise definition of jets is required. The algorithm that relates a jet of hadrons to

partons must satisfy,

- infrared safety - the presence or absence of additional soft particles between two particles

belonging to the same jet should not affect the recombination of these two particles into a jet.

Generally, any soft particles not coming from the fragmentation of a hard scattered parton

should not affect the number of produced jets;

- collinear safety - a jet should be reconstructed independently whether the transverse mo-

mentum is carried by one particle, or if that particle is split into two collinear particles;

- order independence - the algorithm should be applicable at parton- hadron- or detector-

level, and lead to the same origin of the jet.

An illustration of infrared and collinear sensitivity in jet-finding is given in figure 5.1.

(a)

(b)

Figure 5.1. An illustration of infrared (a) and collinear (b) sensitivity in jet finding.

There are many jet algorithms which have been proposed over the years. These include fixed

sized cone algorithms as well as sequential recombination algorithms (cluster algorithms), which

are based on event shape analysis [134]. The term cone algorithm is applied to a wide range of jet

algorithms which broadly aim at maximizing Et inside a geometric circle in the η,φ coordinates

space. The radius of the circle, R, is a key parameter of the algorithm. On the other hand, clustering

algorithms are based upon pair-wise clustering of the input constituents. In general, an algorithm

defines a distance measure between constituents, as well as some condition upon which clustering

into a jet should be terminated.

37

5. Jet reconstruction and calibration

The anti-k⊥

[135], k⊥

[136] and Cambridge/Aachen [137, 138] clustering algorithms are used in

this analysis. The clustering algorithms begin by computing for each jet constituent, the quantity

diB and for each pair of constituents the quantity dij, defined as

diB

= k

2p

Ti ,

(5.1)

dij

= min(k

2p

Ti ,k

2p

Tj)

(∆R)2

ij

R2

,

(5.2)

where

(∆R)2

ij = (yi −yj)2 +(φi −φj)2 .

The variables R and p are constants of the algorithm, and kTi, yi and φi are respectively the transverse

momentum, the rapidity and the azimuth of the ith constituent. The dij parameter, represents the

distance between a pair of jet constituents, while diB is the distance between a given constituent and

the beam.

The algorithm proceeds by comparing for each constituent the different d values. In the case that

the smallest entry is a dij, constituents i and j are combined and the list is remade. If the smallest

entry is diB, this constituent is considered a complete jet and is removed from the list.

The variable R sets the resolution at which jets are resolved from each other as compared to the

beam. For large values of R, the dij are smaller, and thus more merging takes place before jets

are complete. The variable p can also take different values; for the k⊥

algorithm p = 1, for the

Cambridge/Aachen algorithm p = 0, and for the anti-k⊥

algorithm p = −1.

The nominal jet collection used for physics in the analysis uses the anti-k⊥

algorithm with size

parameter, R = 0.6, reconstructed with the FASTJET package [139]. Jets built with the k⊥

algorithm

are also used, but only as part of the jet energy calibration procedure, as discussed in chapter 7.

The Cambridge/Aachen algorithm is used in order to evaluate the systematic uncertainty of said

calibration.

5.2. Inputs to jet reconstruction

5.2.1. Calorimeter jets

The input to calorimeter jets are topological calorimeter clusters (topo-clusters) [126, 140]. Topo-

clusters are groups of calorimeter cells that are designed to follow shower development in the

calorimeter. A topo-cluster is defined as having an energy equal to the energy sum of all the included

calorimeter cells, zero mass and a reconstructed direction calculated from the weighted averages of

the pseudo-rapidities and azimuthal angles of its constituent cells. The weight used in the averaging

is the absolute cell energy. The positions of the cells are relative to the nominal ATLAS coordinate

system.

Clustering algorithm

The topo-cluster algorithm is designed to suppress calorimeter noise. The algorithm starts from a

seed cell, whose signal-to-noise (S/N) ratio is above a threshold, S/N = 4. Cells neighbouring the

seed (or the cluster being formed) that have a signal-to-noise ratio, S/N ≥ 2, are included iteratively.

Finally, all calorimeter cells neighbouring the formed topo-cluster are added.

38

5.2. Inputs to jet reconstruction

The noise is estimated as the absolute value of the energy deposited in the cell, divided by the

RMS of the energy distribution of the cell, measured in events triggered at random bunch crossings.

For data taken during 2010, cell-noise was dominated by electronic noise. Due to the shortened

bunch crossing interval and increased instantaneous luminosity during 2011 data-taking, the noise

increased. An additional component was added, originating from energy deposited in a given cell

from previous collisions, but inside the window of sensitivity of the calorimeters. This added energy

is referred to as pile-up (see chapter 7 for a detailed discussion). Subsequently, different nominal

noise thresholds, σnoise, are used to reconstruct topo-clusters in 2010 and in 2011,

σnoise = 



σelc

noise

(2010)

√(σelc

noise)2

+(σpu

noise)2

(2011)

.

(5.3)

Here σelc

noise

is the electronic noise, and σpu

noise

is the noise from pile-up. On an event-by-event basis,

the magnitude of the latter term may vary, as the pile-up is sensitive to the instantaneous luminosity,

characterized by the average number of interactions per bunch crossing, µ, (see section 7.1.1). A

constant baseline is chosen for cluster reconstruction throughout 2011. This baseline corresponds

to the noise which results from eight additional pile-up interactions, µ = µref = 8, and reflects the

conditions under which the average effects of pile-up in the 2011 data are minimal.

The change in time of the total nominal noise and its dependence on calorimeter pseudo-rapidity

is observed by comparing figures 5.2a and 5.2b. The former shows the energy-equivalent cell-noise

|η|

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

=0) (MeV)

μ

T

o

t. n

o

is

e

(

1

10

2

10

3

10

4

10

FCal1

FCal2

FCal3

HEC1

HEC2

HEC3

HEC4

PS

EM1

EM2

EM3

Tile1

Tile2

Tile3

(a)

|η|

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

=50 ns, DB) (MeV)

A

=8,

μ

T

o

t. n

o

is

e

(

1

10

2

10

3

10

4

10

FCal1

FCal2

FCal3

HEC1

HEC2

HEC3

HEC4

PS

EM1

EM2

EM3

Tile1

Tile2

Tile3

(b)

Figure 5.2. The energy-equivalent cell-noise in the ATLAS calorimeters at the electromagnetic

(EM) scale, as a function of pseudo-rapidity, η. The magnitude of the noise represents the config-

urations used for cluster reconstruction in 2010 (a) and in 2011 (b). These respectively use µ = 0

and µ = 8 as baseline values for the average number of interactions. The various data points indi-

cate the noise in different calorimeter elements; the pre-sampler (PS); the first three layers of the

EM calorimeter (EM1, EM2 and EM3); the first three layers of the Tile calorimeter (Tile1, Tile2

and Tile3); the first three layers of the forward FCal calorimeter (FCal1, FCal2 and FCal3); the

first four layers of the hadronic end-cap calorimeter (HEC1, HEC2 and HEC3). (Figures are taken

from [141].)

39

5. Jet reconstruction and calibration

in different calorimeter regions in 2010, representing the baseline electronic noise; the latter shows

the respective cell-noise in 2011, for which pile-up contributes as well. In most calorimeter regions,

the pile-up induced noise is smaller or of the same magnitude as the electronic noise. The exception

is the forward calorimeters, where σpu

noise ≫ σelec

noise

.

The topo-cluster algorithm also includes a splitting step in order to optimise the separation of

showers from different close-by particles: All cells in a topo-cluster are searched for local maxima

in terms of energy content with a threshold of 500 MeV. This means that the selected calorimeter

cell has to be more energetic than any of its neighbours. The local maxima are then used as seeds for

a new iteration of topological clustering, which splits the original cluster into more topo-clusters.

Energy scale of clusters

Topo-clusters are initially constructed from energy at the electromagnetic (EM) scale. The EM

scale correctly reconstructs the energy deposited by particles in an electromagnetic shower in the

calorimeter. This energy scale is established using test-beam measurements for electrons in the

barrel [121,142–145] and the endcap calorimeters [126,127]. The absolute calorimeter response

to energy deposited via electromagnetic processes was validated in the hadronic calorimeters us-

ing muons, both from test-beams [121,146] and produced in-situ by cosmic rays [147]. The energy

scale of the electromagnetic calorimeters is corrected using the invariant mass of Z bosons produced

in-situ in proton-proton collisions (Z → e

+

e− events) [148].

An additional calibration step may also be used, where so called Local Calibration Weights (LCW)

are applied to topo-clusters, bringing the clusters to the LCW calibration scale [149]. Starting at the

EM scale, topo-clusters are classified as either EM-like or hadron-like, using cluster shape moments.

Clusters classified as EM-like are kept at the original energy scale. On the other hand, hadron-like

topo-clusters are subject to recalibration. In the first step this entails a cell weighting procedure,

which aims to compensate for the lower response of the calorimeter to hadronic deposits. Next,

out-of-cluster corrections are applied. These try to account for lost energy deposited in calorimeter

cells outside of reconstructed clusters. Finally, dead material corrections are applied, accounting for

energy deposits outside of active calorimeter volumes, e.g., in the cryostat, the magnetic coil and

calorimeter inter-modular cracks.

Calorimeter jets are built using either EM or LCW scale topo-clusters. In either case, further cor-

rections need to be applied to calibrate jets to the particle-level. This final step is referred to as the

jet energy scale (JES) calibration.

5.2.2. Other types of jets

Track jets

Jets can also be built using inner detector tracks as inputs to the jet finding algorithm. Track-based

jets, called track jets, are reconstructed using the anti-k⊥

algorithm with size parameter, R = 0.6.

Track jets are required to be composed of at least two tracks; tracks are required to have transverse

momentum above 500 MeV, at least one pixel detector hit, at least six hits in the silicon-strip detec-

tor, and impact parameters in the transverse plane and in the z-direction, both smaller than 1.5 mm.

In order to be fully contained in the tracking region, track jets are required to have |η| < 1.9. Track

jets are also required to have transverse momentum larger than 4 GeV.

40

5.3. Jet energy calibration

Truth jets

Monte Carlo simulated particle jets are referred to as truth jets. They are defined as those built

from stable interacting particles with a lifetime longer than 10 ps in the Monte Carlo event record

(excluding muons and neutrinos) that have not been passed through the simulation of the ATLAS

detector. Truth jets are built using the anti-k⊥

algorithm with size parameter R = 0.6, and are

required to have transverse momentum larger than 10 GeV.

5.3. Jet energy calibration

The jet energy calibration relates the jet energy measured with the ATLAS calorimeter to the true

energy of the corresponding jet of stable particles entering the detector. The jet calibration corrects

for the following detector effects that affect the jet energy measurement;

- calorimeter non-compensation - partial measurement of the energy deposited by

hadrons;

- dead material - energy losses in inactive regions of the detector;

- leakage - energy of particles reaching outside the calorimeters;

- out of calorimeter jet cone - energy deposits of particles which are not included in the

reconstructed jet, but were part of the corresponding truth jet and entered the detector;

- noise thresholds and particle reconstruction efficiency - signal losses in calorimeter

clustering and jet reconstruction.

Jets reconstructed in the calorimeter system are formed from calorimeter energy depositions re-

constructed at either the EM or the LCW scale, as described above. The correction for the lower

response to hadrons is solely based on the topology of the energy depositions observed in the

calorimeter. The measured jet energy is corrected on average, using:

E

calib = E

det ×Fcalib

S

(E

det

S ,η) , with E

det

S = ES −OS

fst , for S = EM , LCW .

(5.4)

Here ES is the calorimeter energy measured at the EM or LCW scales, denoted collectively by S. The

variable Ecalib is the hadron-level calibrated jet energy, and Fcalib

S

is a calibration function. The

latter depends on the measured jet energy, and is evaluated in small jet-pseudo-rapidity, η, regions.

The baseline PYTHIA MC samples described in chapter 4 are used to derive Fcalib

S

; the procedure is

explained in the following.

The variable Edet

S denotes the EM- or LCW-level energy after the contribution of pile-up (additional

multiple proton-proton interactions) has been subtracted. The pile-up energy is expressed through

the function OS

fst

, called the jet offset correction. The offset correction in 2010 and in 2011 depends

on the number of pp collisions which occur within the same bunch crossing as the collision of

interest. This is referred to as in-time pile-up and is estimated by the number of primary vertices,

NPV, in a given event. In addition, OS

fst

exhibits a strong rapidity dependence, due to the varying

calorimeter response in η, and to the rapidity distribution of the pile-up interactions. The offset

correction is parametrized separately for EM and for LCW jets, based in each case on jet constituents

which are calibrated in the respective energy scale.

The offset correction performs well for data taken in 2010 [150]. For the case of the 2011 data,

the nature of pile-up becomes more complicated, as interactions in bunch crossings proceeding

41

5. Jet reconstruction and calibration

and following the event of interest affect the signal in the calorimeter; this effect is called out-

of-time pile-up. The offset correction is modified in 2011 in order to account for this effect, but

the performance is nonetheless degraded with regard to 2010. A detailed discussion of out-of-time

pile-up and the exact functional form of OS

fst

are given in section 7.1. An alternative to the offset

correction using a data-driven method, referred to as the jet area/median method, is introduced in

chapter 7. The offset correction is used for the calibration of jets in the 2010 data in this analysis;

the jet median approach is used as the nominal pile-up correction for the 2011 data.

The two calibrations schemes, which respectively use jets constructed from either EM- or LCW-

calibrated topo-clusters, are referred to as EM+JES and LCW+JES. For data taken during 2010, only

the simpler EM+JES calibration is available for this analysis. For the 2011 data, the nominal jet

collection is calibrated using LCW+JES. The calibration schemes include several steps, described in

the following:

- jet origin correction;

- jet energy correction;

- jet pseudo-rapidity correction;

- residual in-situ calibration (2011 data only).

Jet origin correction

Calorimeter jets are reconstructed using the geometrical center of the ATLAS detector as reference

to calculate the direction of jets and their constituents. The jet four-momentum is corrected for

each event such that the direction of each topo-cluster points back to the primary (highest-pt) recon-

structed vertex. The kinematic observables of each topo-cluster are recalculated using the vector

from the primary vertex to the topo-cluster centroid as its direction. The raw jet four-momentum is

thereafter redefined as the vector sum of the topo-cluster four-momenta. This correction improves

the angular resolution and results in a small improvement (< 1%) in the jet pt response. The jet

energy is unaffected.

Jet energy correction

The principle step of the EM+JES and of the LCW+JES jet calibrations restores the reconstructed

jet energy to the energy of the corresponding MC truth jet. Since pile-up effects have already been

accounted for, the MC samples used to derive the calibration do not include multiple pp interactions.

The calibration is derived using isolated calorimeter jets that have a matching isolated truth jet

within ∆R = 0.3. Distance is defined in η,φ space as

∆R = √(φrec −φtruth)

2

+(ηrec −ηtruth)

2

,

(5.5)

where φtruth (ηtruth) and φrec (ηrec) are respectively the azimuthal angles (pseudo-rapidities) of truth

and reconstructed jets. An isolated jet is defined as one having no other jet with pt > 7 GeV within

∆R = 2.5·R, where R is the size parameter of the jet algorithm.

The final jet energy scale calibration is parametrised as a function of detector-level pseudo-

rapidity, ηdet,1 and of jet energy at the detector-level energy scales, ES. In the following, as above,

S stands for either the EM or the LCW energy scales. The detector-level jet energy-response,

RS = ES/Etruth ,

(5.6)

1 Here, pseudo-rapidity refers to that of the original reconstructed jet before the origin correction.

42

5.3. Jet energy calibration

is measured for each pair of calorimeter and truth jets in bins of truth jet energy, Etruth, and of

ηdet. For each (Etruth,ηdet)-bin, the averaged jet-response, 〈RS〉, is defined as the peak position of

a Gaussian fit to the ES/Etruth distribution. In addition, in the same (Etruth,ηdet)-bin, the average jet

energy-response, 〈ES〉, is derived from the mean of the distribution of ES.

For a given ηdet-bin, k, the jet energy-response calibration function, F

(k)

calib

(ES), is obtained using

a fit of the (〈ES〉j ,〈RS〉j) values for each Etruth-bin j. The fitting function is parameterised as

F

(k)

calib(ES) =

Nmax

∑

i=0

ai (lnES)

(i)

,

(5.7)

where ai are free parameters, and Nmax is chosen between 1 and 6, depending on the goodness of

the fit.

The final jet energy scale correction that relates the measured calorimeter jet energy to the true

energy, is then defined as Fcalib

S

in the following:

ES+JES =

ES

FS

calib

(ES)|k

ηdet

= ES ×Fcalib

S

,

(5.8)

where Fcalib(ES)|k

ηdet

is the jet-response calibration function for the relevant ηdet-bin k.

As an example, the average jet energy scale correction for EM jets in 2010, 〈Fcalib

EM 〉, is shown

as a function of calibrated jet pt for three η-intervals in figure 5.3a. The response of 2010 EM+JES

jets, REM, is shown for various energy and ηdet-bins in figure 5.3b. As the transverse momentum

of jets increases, the response and subsequent JES correction factor decrease. The corrections is

also highly rapidity-dependant, due in part to the changing structure of calorimeter elements; in

transition regions of the calorimeter (around |η| = 1.5,3.3) the response is significantly higher, due

to energy loss in dead material or through leakage. Larger correction factors are therefore required

in these regions. Corresponding figures for 2011 LCW+JES jets may be found in [151].

Jet pseudo-rapidity correction

After the jet origin and energy corrections, the origin-corrected pseudo-rapidity of jets is further cor-

rected for a bias due to poorly instrumented regions of the calorimeter. In these regions topo-clusters

are reconstructed with a lower energy with respect to better instrumented regions (see figure 5.3b).

This causes the jet direction to be biased towards the better instrumented calorimeter regions.

The η-correction is derived as the average difference between the pseudo-rapidity of recon-

structed jets and their truth jet counterparts. It is parametrized according to the energy and detector-

level pseudo-rapidity of reconstructed jets, ∆η = ηtruth − ηorigin. The correction is very small

(< 0.01) for most regions of the calorimeter, and up to five times larger in the transition regions.

Residual in-situ calibration (2011 data only)

Following the pseudo-rapidity correction, the transverse momentum of jets in the 2011 data is com-

pared to that in MC using in-situ techniques. The latter exploit the balance between the transverse

momentum of a jet, pt, and that of a reference object, pref

t

[151]. The ratio,

fin-situ = 〈

pt/pref

t 〉data

〈pt/pref

t 〉MC

,

(5.9)

43

5. Jet reconstruction and calibration

[GeV]

jet

T

p

20 30

2

10

2

10

×

2

3

10

3

10

×

2

A

v

e

ra

g

e

J

E

S

c

o

rre

c

tio

n

1

1.2

1.4

1.6

1.8

2

| < 0.8

η|

≤

0.3

| < 2.8

η|

≤

2.1

| < 4.4

η|

≤

3.6

= 0.6, EM+JES

R

t

Anti-k

ATLAS simulation

(a)

|

det

η|

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Jet response at EM scale

0.4

0.5

0.6

0.7

0.8

0.9

1

E = 30 GeV

E = 60 GeV

E = 110 GeV

E = 400 GeV

E = 2000 GeV

Forward

Endcap-Forward

Transition

Endcap

Barrel-Endcap

Transition

Barrel

= 0.6, EM+JES

R

t

Anti-k

ATLAS simulation

(b)

Figure 5.3. (a) Average jet energy scale correction for EM+JES jets as a function of the calibrated

jet transverse momentum, p

jet

T

, for three pseudo-rapidity, η, regions, as indicated in the figure.

(b) Average simulated jet-response at the EM scale in bins of detector pseudo-rapidity, ηdet. Different

jet energies, E, are represented (using EM+JES jets) as indicated in the figure. The parallel lines mark

the regions over which the JES correction is evaluated. The inverse of the response shown in each

bin is equal to the average jet energy scale correction.

(Figures are taken from [150].)

called the residual in-situ JES correction, is used on jets measured in data. It is derived using the

following methods:

- the pt of jets within 0.8 < |η| < 4.5 is equalized to the pt of jets within |η| < 0.8, exploiting

the pt balance between central and forward jets in events with only two high pt jets;

- an in-situ JES correction for jets within |η| < 1.2 is derived using the pt of a photon or a

Z boson (decaying to e

+

e− or µ+µ−) as reference;

- events where a system of low-pt jets recoils against a high-pt jet are used to calibrate jets in

the TeV regime. The low- and high-pt jets are required to respectively be within |η| < 2.8 and

|η| < 1.2.

5.4. Jet quality selection

Jets at high transverse momenta produced in pp collisions must be distinguished from background

jets not originating from hard scattering events. The main sources of background are listed in the

following:

- large calorimeter noise;

- beam-gas events, where one proton of the beam collided with the residual gas within the beam

pipe;

- beam-halo events, e.g., caused by interactions in the tertiary collimators in the beam-line far

away from the ATLAS detector;

44

5.5. Systematic uncertainties on the kinematic properties of jets

- cosmic ray muons overlapping in-time with collision events.

These backgrounds are divided into two categories, calorimeter noise and non-collision interactions.

Calorimeter noise - two types of calorimeter noise are addressed, sporadic noise bursts and

coherent noise. Sporadic noise bursts in the hadronic endcap calorimeter (HEC) commonly result

in a single noisy calorimeter cell, which contributes almost all of the energy of a jet. Such jets

are therefore rejected if they have high HEC energy fractions, denoted by fHEC. The signal shape

quality, SHEC, may also be used for rejection, the latter being a measure of the pulse shape of a

calorimeter cell compared to nominal conditions. Due to the capacitive coupling between channels,

neighbouring calorimeter cells around the noise burst have an apparent negative energy, denoted by

Eneg. A hight value of Eneg is therefore used to distinguish jets which originate in noise bursts. On

rare occasions, coherent noise in the electromagnetic calorimeter develops. Fake jets arising from

this background are characterised by a large EM energy fraction, fEM, which is the ratio of the energy

deposited in the EM calorimeter to the total energy. Similar to the case of noise bursts in the HEC, a

large fraction of calorimeter cells exhibit poor signal shape quality, SEM, in such cases.

Non-collision backgrounds - cosmic rays or non-collision interactions are likely to induce

events where jet candidates are not in-time with the beam collision. A cut on the jet-time, tjet, may

therefore be applied to reject such jets. Jet-time is reconstructed from the energy deposition in the

calorimeter by weighting the reconstructed time of calorimeter cells forming the jet with the square

of the cell energy. The calorimeter time is defined with respect to the event time (recorded by the

trigger). A cut on fEM is applied to make sure that jets have some energy deposited in the calorimeter

layer closest to the interaction region, as expected for a jet originating from the nominal interaction

point. Since real jets are expected to have tracks, the fEM cut may be applied together with a cut on

the jet charged fraction, fch, defined as the ratio of the scalar sum of the pt of the tracks associated to

the jet, divided by the jet pt. The jet charged fraction cut is, naturally, only applicable for jets within

the acceptance of the tracker. A cut on the maximum energy fraction in any single calorimeter layer,

fmax, is applied to further reject non-collision background.

Two sets of quality criteria are defined, denoted as Loose and Medium selection. These incorpo-

rate requirements on the rejection variables defined above, and are specified in detail in section A.1,

table A.1. For the 2010 data, the tighter, Medium selection, is required. Since these criteria are not

fully efficient, an efficiency correction is applied to jets. The selection efficiency for jets in two

pseudo-rapidity regions is shown in figure 5.4 for illustration. In general, for transverse momenta

larger than ∼ 50 GeV, the efficiency is above 99% across all rapidities. For lower jet pt, efficiencies

range between 96-98% within |η| < 2.1, and are above 99% otherwise. Further details are available

in [150]. For the 2011 data, due to improved understanding of the calorimeter, a reduced version of

the 2010 Loose selection is used. The efficiency is above 99.8% for jets with pt > 30 GeV across

all rapidities.

5.5. Systematic uncertainties on the kinematic properties

of jets

5.5.1. Jet energy scale uncertainties

The systematic uncertainty associated with the jet energy scale (JES) is the primary source of uncer-

tainty in the analysis. It has been determined for the 2010 and 2011 datasets in [152] and in [151],

45

5. Jet reconstruction and calibration

[GeV]

jet

T

p

30 40 50

2

10

2

10

×

2

Jet quality selection efficiency

0.95

0.96

0.97

0.98

0.99

1

1.01

ATLAS

= 7 TeV, Data 2010

s

R=0.6

t

anti-k

| < 0.3

η|

Loose selection

Medium selection

(a)

[GeV]

jet

T

p

30 40 50

2

10

2

10

×

2

Jet quality selection efficiency

0.95

0.96

0.97

0.98

0.99

1

1.01

ATLAS

= 7 TeV, Data 2010

s

R=0.6

t

anti-k

| < 0.8

η|

≤

0.3

Loose selection

Medium selection

(b)

Figure 5.4. Jet quality selection efficiency for anti-k⊥

jets with size parameter, R = 0.6, measured

as a function of the transverse momentum of jets, p

jet

T

, in two pseudo-rapidity, η, regions. The

Loose and Medium selection criteria used for data taken during 2010 are represented, as indicated

in the figures. (Figures taken from [150].)

using integrated luminosities of 38 pb−1 and 4.7 fb−1, respectively. In total seven sources of uncer-

tainty on the jet energy scale are considered in 2010 and thirteen in 2011.

The sources of uncertainty on the 2010 measurement are the following:

- single hadron response - uncertainty associated with the response of a single particle

entering the calorimeter. Discrepancies may arise due to the limited knowledge of the exact

detector geometry; due to the presence of additional dead material; and due to the modelling

of the exact way particles interact in the detector;

- cluster thresholds - uncertainty associated with the thresholds for reconstructing topo-

clusters. The clustering algorithm is based on the signal-to-noise ratio of calorimeter cells.

Discrepancies between the simulated noise and the real noise, or changes in time of the noise

in data, can lead to differences in the cluster shapes and to the presence of fake topo-clusters;

- Perugia 2010 and Alpgen+Herwig+Jimmy - uncertainty associated with the mod-

elling of fragmentation and the underlying event, or with other choices in the event modelling

of the MC generator. The response predicted by the nominal PYTHIA generator are compared

to the PYTHIA Perugia 2010 tune and to ALPGEN, coupled to HERWIG and JIMMY;

- intercalibration - uncertainty associated with the rapidity-intercalibration method, in which

dijet events are used in-situ to measure the response in two η regions in the calorimeter. The

measurement is done in different rapidity intervals simultaneously, by minimizing a response

matrix. The uncertainty is estimated by comparing the response with that measured in-situ in

events in which one of the jets is constrained to be central;

- relative non-closure - uncertainty associated with the non-closure of the energy of jets in

MC following the JES calibration;

- in-time pile-up - uncertainty associated with the simulation and subtraction of in-time

pile-up.

46

5.5. Systematic uncertainties on the kinematic properties of jets

For the LCW+JES calibration scheme, which is used in this analysis for the 2011 dataset, the base-

line JES uncertainty is estimated using a combination of in-situ techniques. In total, the different

sources of uncertainty coming from the in-situ techniques in 2011 data amount to 60 components.

These are combined, assuming they are fully correlated in pt and η, into six effective nuisance pa-

rameters (ENP). The combination procedure involves diagonalizing the covariance matrix of the

JES correction factors and selecting the five eigenvectors that have the largest corresponding eigen-

values. An additional sixth effective parameter represents all residual sources of uncertainty.

The sources of uncertainty on the 2011 measurement are the following:

- ENP 1-6 - uncertainty associated with the six effective nuisance parameters, which combine

the 60 components of the in-situ methods used to estimate the JES uncertainty in 2011;

- intercalibration, single hadron response and relative non-closure - same as for

2010;

- close-by jets - uncertainty associated with event topologies in which two jets are recon-

structed in close proximity;

- in-time PU, out-of-time PU and PU pt - uncertainty associated with the simulation

and subtraction of in- and out-of-time pile-up.

The final fractional JES uncertainties for the 2010 and the 2011 data are compared in figure 5.5 as

a function of the pt of jets in the central region. The uncertainty on the 2011 data is much improved

[GeV]

jet

T

p

20 30 40

2

10

2

10

×

2

3

10

3

10

×

2

Fractional JES uncertainty

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

ATLAS Internal

= 0.6, EM+JES

R

t

Anti-k

= 0.5

| η |

2011

2010

Before collisions

Baseline JES uncertainties

(a)

[GeV]

jet

T

p

20 30 40

2

10

2

10

×

2

3

10

3

10

×

2

Fractional JES uncertainty

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

ATLAS Internal

= 0.6, LCW+JES

R

t

Anti-k

= 0.5

| η |

2011

2010

Baseline JES uncertainties

(b)

Figure 5.5. Fractional jet energy scale systematic uncertainty as a function of the transverse

momentum of jets, p

jet

T, for jets with pseudo-rapidity, |η| = 0.5, calibrated under EM+JES (a) and

LCW+JES (b). As indicated in the figures, estimates of the uncertainty before beginning of collisions

in the LHC, for data taken during 2010 and for data taken during 2011, are compared. The 2011 un-

certainty presented here includes only the in-situ components of the systematic uncertainty. (Figure

are taken from [151].)

compared to 2010. This is due both to the increase in 2011 in the statistics which are used to derive

uncertainty, and to the use of sophisticated in-situ techniques. For EM+JES jet in 2010 the total

47

5. Jet reconstruction and calibration

uncertainty is roughly 2% for jets with 40 < pt < 2·103 GeV, increasing up to roughly twice that

for lower or higher transverse momenta. For LCW+JES jets in 2011, the uncertainty is roughly 1% in

the respective mid-pt range, increasing up to 2% between 20 and 40 GeV and up to 4% between 0.5

and 2 TeV.

5.5.2. Jet energy and angular resolution

In addition to the jet energy scale, which is the primary source of uncertainty for most measurements

involving jets, the jet energy and angular resolution must also be taken into account. These are

addressed in the following.

Jet energy resolution

The baseline parameterization of the jet energy resolution in MC is derived using the nominal

PYTHIA MC simulation for jets with transverse momentum, 30 < pt < 500 GeV [153]. In-situ mea-

surements are also made for jets with rapidities, |y| < 2.8 and pt > 20 GeV. In this kinematic range,

the comparison of the resolution measured with the in-situ techniques between the data and the MC

shows agreement better than 10%. The uncertainty on the jet energy resolution for each rapidity

region is assigned from the weighted average of the systematic errors on the relative difference be-

tween the data and the MC, and is flat as a function of pt. Outside the kinematic range of in-situ

measurements, the MC parameterization is kept but the uncertainty is conservatively increased.

In order to illustrate the behaviour of the transverse momentum resolution in MC, the relative

transverse momentum offset between the pt of a truth jet, ptruth

t

, and that of the corresponding

matched reconstructed jet, prec

t , is defined as

Opt =

prec

t

− ptruth

t

ptruth

t

.

(5.10)

Matching is performed by selecting a reconstructed jet within distance ∆Rmatch = 0.5·R of the truth

jet, where R = 0.6 is the size parameter of the jet. In order to keep the sample pure, isolation of the

matched truth jets from their counterparts within ∆Riso = 2·R is also imposed.

The average relative transverse momentum offset represents the fractional bias in pt of jets. The

relative pt resolution is parametrized by the width of the distribution of Opt, denoted by σ (Opt ).

The dependence of the average Opt and of σ (Opt) on truth jet pt is shown in figure 5.6 for jets in dif-

ferent pseudo-rapidity regions in MC10 and in MC11. At pt < 20 GeV in 2010 and at pt < 30 GeV

in 2011, Opt changes sign. These transverse momentum values mark the corresponding thresholds

in either year for which the energy calibration of jets becomes valid. The relative bias in transverse

momentum is lower in absolute value than 1% (6%) for jets with pt < 100 GeV within |η| < 2.1

(|η| < 4.5) in 2010. In 2011, the bias is lower than 2% for central jets within |η| < 2.8. For higher

values of transverse momentum the relative bias decreases, becoming negligible for non-central

jets in 2010 above 200 GeV and for central jets in 2011 above 100 GeV. The relative resolution,

σ (Opt), is also rapidity- and momentum-dependent. For the various η-regions the relative reso-

lution decreases from 21% in 2010 (and 27% in 2011) for jets with pt = 20 GeV, down to 4-6%

for jets with pt > 300 GeV. Both Opt and σ (Opt ) are generally higher for low-pt jets in MC11

compared to MC10, due to the increase in pile-up in 2011.

48

5.5. Systematic uncertainties on the kinematic properties of jets

[GeV]

truth

t

p

0 100 200 300 400 500 600 700

> tp

<O

-0.05

0

0.05

0.1

0.15

MC10 Pythia

R=0.6 cluster jets

t

anti-k

Internal

ATLAS

| < 0.3

truth

η

0.0 < |

| < 0.8

truth

η

0.3 < |

| < 1.2

truth

η

0.8 < |

| < 2.1

truth

η

1.2 < |

| < 2.8

truth

η

2.1 < |

| < 3.2

truth

η

2.8 < |

| < 3.6

truth

η

3.2 < |

| < 4.5

truth

η

3.6 < |

(a)

[GeV]

truth

t

p

0 100 200 300 400 500 600 700

> tp

<O

-0.05

0

0.05

0.1

0.15

MC11 Pythia

R=0.6 cluster jets

t

anti-k

Internal

ATLAS

| < 0.3

truth

η

0.0 < |

| < 0.8

truth

η

0.3 < |

| < 1.2

truth

η

0.8 < |

| < 2.1

truth

η

1.2 < |

| < 2.8

truth

η

2.1 < |

(b)

[GeV]

truth

t

p

0

0.2

0.4

0.6

0.8

1

3

10

×

) tp

(O

σ

0

0.05

0.1

0.15

0.2

0.25

0.3

MC10 Pythia

R=0.6 cluster jets

t

anti-k

Internal

ATLAS

| < 0.3

truth

η

0.0 < |

| < 0.8

truth

η

0.3 < |

| < 1.2

truth

η

0.8 < |

| < 2.1

truth

η

1.2 < |

| < 2.8

truth

η

2.1 < |

| < 3.2

truth

η

2.8 < |

| < 3.6

truth

η

3.2 < |

| < 4.5

truth

η

3.6 < |

(c)

[GeV]

truth

t

p

0

0.2

0.4

0.6

0.8

1

3

10

×

) tp

(O

σ

0

0.05

0.1

0.15

0.2

0.25

0.3

MC11 Pythia

R=0.6 cluster jets

t

anti-k

Internal

ATLAS

| < 0.3

truth

η

0.0 < |

| < 0.8

truth

η

0.3 < |

| < 1.2

truth

η

0.8 < |

| < 2.1

truth

η

1.2 < |

| < 2.8

truth

η

2.1 < |

(d)

Figure 5.6. Dependence of the average relative transverse momentum offset, < Opt >, ((a) and

(b)) and of the width of the distribution of the relative offset, σ(Opt), ((c) and (d)) on the transverse

momentum of truth jets, ptruth

t

. Several regions of truth jet pseudo-rapidity, ηtruth, are presented for

jets in MC10 and in MC11, as indicated in the figures.

49

5. Jet reconstruction and calibration

Jet angular resolution

In addition to the pt resolution, the rapidity and azimuthal resolutions are used in section 8.5 to

estimated a component of the systematic uncertainty associated with the dijet mass measurement.

A parametrization of the angular resolutions for different jet transverse momenta and rapidities is

derived from the following.

The pseudo-rapidity offset and the azimuthal offset of jets are respectively defined in MC as

Oη = ηrec −ηtruth

and Oφ = φrec −φtruth ,

(5.11)

by matching truth and reconstructed jets, as discussed with regard to equation (5.10). Here, as

before, ηtruth (φtruth) and ηrec (φrec) are respectively the pseudo-rapidities (azimuthal angles) of

truth and reconstructed jets. The parameters Oη and Oφ represents the angular bias of jets. The

width of the distributions of Oη and Oφ represent the angular resolution of jets, denoted respectively

as σ (Oη) and σ (Oφ ).

The dependence of the average angular offset parameters and of the angular resolution parameters

on truth jet pt is shown in figures 5.7 - 5.8 for jets in different pseudo-rapidity regions in MC10 and

in MC11. The pseudo-rapidity offset is smaller than 3·10

−3 in all η regions for jets with low pt

in 2010, becoming insignificant for transverse momentum values above 200 GeV. For jets in 2011

a small bias of roughly 2 ·10

−3 is observed for all η regions. The η resolution is roughly 0.04

for jets with pt = 20 GeV in MC10 and for jets with pt = 30 GeV in MC11. For jets with higher

transverse momenta, the resolution decreases, reaching values smaller than 3·10

−3 above 200 GeV.

The slightly degraded performance for jets in 2011 compared to 2010 is due the increase in pile-up.

The azimuthal offset and resolution follow similar trends as do the pseudo-rapidity offset and

resolution parameters. The magnitude of the offset is smaller than 3·10

−3 for low pt jets, becoming

insignificant (consistent with zero within errors) above ∼ 100 GeV. The azimuthal resolution also

improves with growing jet pt, decreasing from 0.07 for jets with pt = 20 GeV (pt = 30 GeV) to

roughly 0.02 above 200 GeV for jets in MC10 (MC11).

50

5.5. Systematic uncertainties on the kinematic properties of jets

[GeV]

truth

t

p

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

3

10

×

>

η

<O

-4

-3

-2

-1

0

1

2

3

4

-3

10

×

MC10 Pythia

R=0.6 cluster jets

t

anti-k

Internal

ATLAS

| < 0.3

truth

η

0.0 < |

| < 0.8

truth

η

0.3 < |

| < 1.2

truth

η

0.8 < |

| < 2.1

truth

η

1.2 < |

| < 2.8

truth